Sharygin 2005-22 805p

geometry problems
with aops links in the names from
Geometrical Olympiad in Honor of I. F. Sharygin (also known as Sharygin Geometry Olympiad)

Sharygin 2022 Correspondence + Final Round links (aops)

Sharygin 2021 Correspondence + Final Round links (aops)

Sharygin Geometry Olympiad 2020 Correspondence Round Official Solutions (pdf)

Sharygin 2020 final round has been cancelled

Sharygin Geometry Olympiad 2020 Correspondence Round problems (pdf)

Sharygin Geometry Olympiad 2020 Correspondence Round links (aops)

juniors finals collected insides aops: grade VIII - grade IX

below are years 2005   - 2019, 2021-22: First  & Final  Round, year 2020: First  Round only

This Olympiad started in 2005 and has two rounds (first and final). Since 2007 problems are also published in English (besides 2010 final). In the following pdfs are in the English one, everything that was published in English, and the Russian one, everything missing from the english collection. The only problems without solution, not even in Russian are the final round in 2005.

The first round is officially called as correspodence.

Sharygin Geometry Olympiad 2020 Correspondence Round (pdf)
Sharygin Geometry Olympiad 2020 Correspondence Round (aops)

Sharygin Geometry Olympiad 2005-06 EN in pdf

Sharygin Geometry Olympiad 2007-19 EN in pdf

Sharygin 2010 final in English  inside aops  here
Sharygin aops collections: 2005 , 2006

collecting in English :

1st round: 2007- 2019 (solutions in 2009-19)
final round: 2007- 2010, 2011-2018 (solutions in 2009-2018)

Sharygin Geometry Olympiad 2005-06 EN in pdf

Sharygin Geometry Olympiad 2005-06 RU in pdf

collecting in Russian :

1st round: problems 2005- 06 (solutions in 2005-08)
Final Round: (solutions in 2005-08)

To be more exact, the Russian pdf collects all the problems and all solutions that are not contained in the English pdf.

2004-2005 First Round

collected inside aops here

2005 Sharygin Geometry Olympiad First Round p1 grade 8

The chords $AC$ and $BD$ of the circle intersect at point $P$. The perpendiculars to $AC$ and $BD$ at points $C$ and $D$, respectively, intersect at point $Q$. Prove that the lines $AB$ and $PQ$ are perpendicular

2005 Sharygin Geometry Olympiad First Round p2 grades 8-9 

Cut a cross made up of five identical squares into three polygons, equal in area and perimeter.

2005 Sharygin Geometry Olympiad First Round p3 grades 8-9 

Given a circle and a point $K$ inside it. An arbitrary circle equal to the given one and passing through the point $K$ has a common chord with the given circle. Find the geometric locus of the midpoints of these chords.

2005 Sharygin Geometry Olympiad First Round p4 grades 9-11 

At what smallest $n$ is there a convex $n$-gon for which the sines of all angles are equal and the lengths of all sides are different?

2005 Sharygin Geometry Olympiad First Round p5 grades 8-10 

There are two parallel lines $p_1$ and $p_2$. Points $A$ and $B$ lie on $p_1$, and $C$ on $p_2$. We will move the segment $BC$ parallel to itself and consider all the triangles $AB'C '$ thus obtained. Find the locus of the points in these triangles:
a) points of intersection of heights,
b) the intersection points of the medians,
c) the centers of the circumscribed circles.

2005 Sharygin Geometry Olympiad First Round p6 grades 10-11 

Side $AB$ of triangle $ABC$ was divided into $n$ equal parts (dividing points $B_0 = A, B_1, B_2, ..., B_n = B$), and side $AC$ of this triangle was divided into $(n + 1)$ equal parts (dividing points $C_0 = A, C_1, C_2, ..., C_{n+1} = C$). Colored are the triangles $C_iB_iC_{i+1}$ (where $i = 1,2, ..., n$). What part of the area of the triangle is painted over?

2005 Sharygin Geometry Olympiad First Round p7 grades 8-9 

Two circles with radii $1$ and $2$ have a common center at the point $O$. The vertex $A$ of the regular triangle $ABC$ lies on the larger circle, and the middpoint of the base $CD$ lies on the smaller one. What can the angle $BOC$ be equal to?

2005 Sharygin Geometry Olympiad First Round p8 grades 8-9 

Around the convex quadrilateral $ABCD$, three rectangles are circumscribed . It is known that two of these rectangles are squares. Is it true that the third one is necessarily a square?

(A rectangle is circumscribed around the quadrilateral $ABCD$ if there is one vertex $ABCD$ on each side of the rectangle).

2005 Sharygin Geometry Olympiad First Round p9 grade 9

Let $O$ be the center of a regular triangle $ABC$. From an arbitrary point $P$ of the plane, the perpendiculars were drawn on the sides of the triangle. Let $M$ denote the intersection point of the medians of the triangle , having vertices the bases of the perpendiculars. Prove that $M$ is the midpoint of the segment $PO$.

2005 Sharygin Geometry Olympiad First Round p10 grades 8-9 

Cut the non-equilateral triangle into four similar triangles, among which not all are the same.

2005 Sharygin Geometry Olympiad First Round p11 grades 8-10 

The square was cut into $n^2$ rectangles with sides $a_i \times b_j, i , j= 1,..., n$. For what is the smallest $n$ in the set $\{a_1, b_1, ..., a_n, b_n\}$ all the numbers can be different?

2005 Sharygin Geometry Olympiad First Round p12 grades 8-9 

Construct a quadrangle along the given sides $a, b, c$, and $d$ and the distance $I$ between the midpoints of its diagonals.

2005 Sharygin Geometry Olympiad First Round p13 grade 9

A triangle $ABC$ and two lines $\ell_1, \ell_2$ are given. Through an arbitrary point $D$ on the side $AB$, a line parallel to $\ell_1$ intersects the $AC$ at point $E$ and a line parallel to $\ell_2$ intersects the $BC$ at point $F$. Construct a point $D$ for which the segment $EF$ has the smallest length.

2005 Sharygin Geometry Olympiad First Round p14 grades 10-11 

Let $P$ be an arbitrary point inside the triangle $ABC$. Let $A_1, B_1$ and $C_1$ denote the intersection points of the straight lines $AP, BP$ and $CP$, respectively, with the sides $BC, CA$ and $AB$. We order the areas of the triangles $AB_1C_1,A_1BC_1,A_1B_1C$. Denote the smaller by $S_1$, the middle by $S_2$, and the larger by $S_3$. Prove that $\sqrt{S_1 S_2} \le S \le  \sqrt{S_2 S_3}$ ,where $S$ is the area of the triangle $A_1B_1S_1$.

2005 Sharygin Geometry Olympiad First Round p15 grade 11

Given a circle centered at the origin. Prove that there is a circle of smaller radius that has no less points with integer coordinates.

2005 Sharygin Geometry Olympiad First Round p16 grades 8-9 

We took a non-equilateral acute-angled triangle and marked $4$ wonderful points in it: the centers of the inscribed and circumscribed circles, the center of gravity (the point of intersection of the medians) and the point of intersection of heights. Then the triangle itself was erased. It turned out that it was impossible to establish which of the centers corresponds to each of the marked points. Find the angles of the triangle.

2005 Sharygin Geometry Olympiad First Round p17 grade 11

A circle is inscribed in the triangle $ABC$ and it's center $I$ and the points of tangency $P, Q, R$ with the sides $BC$, $C A$ and $AB$ are marked, respectively. With a single ruler, build a point $K$ at which the circle passing through the vertices B and $C$ touches (internally) the inscribed circle.

2005 Sharygin Geometry Olympiad First Round p18 grade 10-11

On the plane are three straight lines $\ell_1, \ell_2,\ell_3$, forming a triangle, and the point $O$ is marked, the center of the circumscribed circle of this triangle. For an arbitrary point X of the plane, we denote by $X_i$ the point symmetric to the point X with respect to the line $\ell_i, i = 1,2,3$.
a) Prove that for an arbitrary point $M$ the straight lines connecting the midpoints of the segments $O_1O_2$ and $M_1M_2, O_2O_3$ and $M_2M_3, O_3O_1$ and $M_3M_1$ intersect at one point,
b) where can this intersection point lie?

2005 Sharygin Geometry Olympiad First Round p19 grades 8-11

As you know, the moon revolves around the earth. We assume that the Earth and the Moon are points, and the Moon rotates around the Earth in a circular orbit with a period of one revolution per month.

The flying saucer is in the plane of the lunar orbit. It can be jumped through the Moon and the Earth - from the old place (point $A$), it instantly appears in the new (at point $A '$) so that either the Moon or the Earth is in the middle of segment $AA'$. Between the jumps, the flying saucer hangs motionless in outer space.

1) Determine the minimum number of jumps a flying saucer will need to jump from any point inside the lunar orbit to any other point inside the lunar orbit.

2) Prove that a flying saucer, using an unlimited number of jumps, can jump from any point inside the lunar orbit to any other point inside the lunar orbit for any period of time, for example, in a second.

2005 Sharygin Geometry Olympiad First Round p20 grade 11

Let $I$ be the center of the sphere inscribed in the tetrahedron $ABCD, A ', B', C ', D'$ be the centers of the spheres circumscribed around the tetrahedra $IBCD, ICDA, IDAB, IABC$, respectively. Prove that the sphere circumscribed around $ABCD$ lies entirely inside the circumscribed around $A'B'C'D '$.

2005 Sharygin Geometry Olympiad First Round p21 grades 10-11

The planet Tetraincognito covered by ocean has the shape of a regular tetrahedron with an edge of $900$ km. What area of the ocean will the tsunami' cover $2$ hours after the earthquake with the epicenter in

a) the center of the face,

b) the middle of the rib,

if the tsunami propagation speed is $300$ km / h?

2005 Sharygin Geometry Olympiad First Round p22 grades 10-11 

Perpendiculars at their centers of gravity (points of intersection of medians) are restored to the faces of the tetrahedron. Prove that the projections of the three perpendiculars to the fourth face intersect at one point.

2005 Sharygin Geometry Olympiad First Round p23 grades 10-11 

Envelop the cube in one layer with five convex pentagons of equal areas.

2005 Sharygin Geometry Olympiad First Round p24 grades 10-11 

A triangle is given, all the angles of which are smaller than $\phi$, where $\phi <2\pi / 3$. Prove that in space there is a point from which all sides of the triangle are visible at an angle $\phi$.

2004-2005  Final Round

collected inside aops here

grade 9

2005 Sharygin Geometry Olympiad Finals 9.1

The quadrangle $ABCD$ is inscribed in a circle whose center $O$ lies inside it. Prove that if $\angle BAO = \angle DAC$, then the diagonals of the quadrilateral are perpendicular.

2005 Sharygin Geometry Olympiad Finals 9.2

Find all isosceles triangles that cannot be cut into three isosceles triangles with the same sides.

2005 Sharygin Geometry Olympiad Finals 9.3

Given a circle and points $A, B$ on it. Draw the set of midpoints of the segments, one of the ends of which lies on one of the arcs $AB$, and the other on the second.

2005 Sharygin Geometry Olympiad Finals 9.4

Let $P$ be the intersection point of the diagonals of the quadrangle $ABCD$, $M$ the intersection point of the lines connecting the midpoints of its opposite sides, $O$ the intersection point of the perpendicular bisectors of the diagonals, $H$ the intersection point of the lines connecting the orthocenters of the triangles $APD$ and $BCP$, $APB$ and $CPD$. Prove that $M$ is the midpoint of $OH$.

2005 Sharygin Geometry Olympiad Finals 9.5

It is given that for no side of the triangle from the height drawn to it, the bisector and the median it is impossible to make a triangle. Prove that one of the angles of the triangle is greater than $135 ^ o$

grade 10

2005 Sharygin Geometry Olympiad Finals 10.1

A convex quadrangle without parallel sides is given. For each triple of its vertices, a point is constructed that supplements this triple to a parallelogram, one of the diagonals of which coincides with the diagonal of the quadrangle. Prove that of the four points constructed, exactly one lies inside the original quadrangle.

2005 Sharygin Geometry Olympiad Finals 10.2

A triangle can be cut into three similar triangles. Prove that it can be cut into any number of triangles similar to each other.

2005 Sharygin Geometry Olympiad Finals 10.3

Two parallel chords $AB$ and $CD$ are drawn in a circle with center $O$.

Circles with diameters $AB$ and $CD$ intersect at point $P$. Prove that the midpoint of the segment $OP$ is equidistant from lines $AB$ and $CD$.

2005 Sharygin Geometry Olympiad Finals 10.4

Two segments $A_1B_1$ and $A_2B_2$ are given on the plane, with $\frac{A_2B_2}{A_1B_1} = k < 1$. On segment $A_1A_2$, point $A_3$ is taken, and on the extension of this segment beyond point $A_2$, point $A_4$ is taken, so $\frac{A_3A_2}{A_3A_1} =\frac{A_4A_2}{A_4A_1}= k$. Similarly, point $B_3$ is taken on segment $B_1B_2$ , and on the extension of this the segment beyond point $B_2$ is point $B_4$, so $\frac{B_3B_2}{B_3B_1} =\frac{B_4B_2}{B_4B_1}= k$. Find the angle between lines $A_3B_3$ and $A_4B_4$.

(Netherlands)

2005 Sharygin Geometry Olympiad Finals 10.5

Two circles of radius $1$ intersect at points $X, Y$, the distance between which is also equal to $1$. From point $C$ of one circle, tangents $CA, CB$ are drawn to the other. Line $CB$ will cross the first circle a second time at point $A'$. Find the distance $AA'$.

2005 Sharygin Geometry Olympiad Finals 10.6

Let $H$ be the orthocenter of triangle $ABC$, $X$ be an arbitrary point. A circle with a diameter of $XH$ intersects lines $AH, BH, CH$ at points $A_1, B_1, C_1$ for the second time, and lines $AX BX, CX$ at points $A_2, B_2, C_2$. Prove that lines A$_1A_2, B_1B_2, C_1C_2$ intersect at one point.

grade 11

2005 Sharygin Geometry Olympiad Finals 11.1

$A_1, B_1, C_1$ are the midpoints of the sides $BC,CA,BA$ respectively of an equilateral triangle $ABC$. Three parallel lines, passing through $A_1, B_1, C_1$ intersect, respectively, lines $B_1C_1, C_1A_1, A_1B_1$ at points $A_2, B_2, C_2$. Prove that the lines $AA_2, BB_2, CC_2$ intersect at one point lying on the circle circumscribed around the triangle $ABC$.

2005 Sharygin Geometry Olympiad Finals 11.2

Convex quadrilateral $ABCD$ is given. Lines $BC$ and $AD$ intersect at point $O$, with $B$ lying on the segment $OC$, and $A$ on the segment $OD$. $I$ is the center of the circle inscribed in the $OAB$ triangle, $J$ is the center of the circle exscribed in the triangle $OCD$ touching the side of $CD$ and the extensions of the other two sides. The perpendicular from the midpoint of the segment $IJ$ on the lines $BC$ and $AD$ intersect the corresponding sides of the quadrilateral (not the extension) at points $X$ and $Y$. Prove that the segment $XY$ divides the perimeter of the quadrilateral$ABCD$ in half, and from all segments with this property and ends on $BC$ and $AD$, segment $XY$ has the smallest length.

2005 Sharygin Geometry Olympiad Finals 11.3

Inside the inscribed quadrilateral $ABCD$ there is a point $K$, the distances from which to the sides $ABCD$ are proportional to these sides. Prove that $K$ is the intersection point of the diagonals of $ABCD$.

2005 Sharygin Geometry Olympiad Finals 11.4

In the triangle $ABC , \angle A = \alpha,  BC = a$. The inscribed circle touches the lines $AB$ and $AC$ at points $M$ and $P$. Find the length of the chord cut by the line $MP$ in a circle with diameter $BC$.

2005 Sharygin Geometry Olympiad Finals 115

11.5 The angle and the point $K$ inside it are given on the plane. Prove that there is a point $M$ with the following property: if an arbitrary line passing through intersects the sides of the corner at points $A$ and $B$, then $MK$ is the bisector of the angle $AMB$.

2005 Sharygin Geometry Olympiad Finals 11.6

The sphere inscribed in the tetrahedron $ABCD$ touches its faces at points $A',B',C',D'$. The segments $AA'$ and $BB'$ intersect, and the point of their intersection lies on the inscribed sphere. Prove that the segments $CC'$ and $DD'$ also intersect on the inscribed sphere.

2005-2006 First Round

collected inside aops here

2006 Sharygin Geometry Olympiad First Round p1 grade 8 
Two straight lines intersecting at an angle of $46^o$ are the axes of symmetry of the figure $F$ on the plane.What is the smallest number of axes of symmetry this figure can have?

2006 Sharygin Geometry Olympiad First Round p2 grades 8,9 
Points $A, B$ move with equal speeds along two equal circles.Prove that the perpendicular bisector of  $AB$ passes through a fixed point.

2006 Sharygin Geometry Olympiad First Round p3 grades 8,9  
The map shows sections of three straight roads connecting the three villages, but the villages themselves are located outside the map. In addition, the fire station located at an equal distance from the three villages is not indicated on the map, although its location is within the map. Is it possible to find this place with the help of a compass and a ruler, if the construction is carried out only within the map?

2006 Sharygin Geometry Olympiad First Round p4 grades 8 (a), 9-11 (b)
a) Given two squares $ABCD$ and $DEFG$, with point $E$ lying on the segment $CD$, and points$F,G$ outside the square $ABCD$. Find the angle between lines  $AE$ and $BF$.
b) Two regular pentagons $OKLMN$ and $OPRST$ are given, and the point $P$ lies on the segment $ON$, and the points $R, S, T$ are outside the pentagon $OKLMN$. Find the angle between straight lines $KP$ and $MS$.

2006 Sharygin Geometry Olympiad First Round p5 grades 8 (a), 9-11 (b) 
a) Fold a $10 \times 10$ square from a $1 \times 118$ rectangular strip.
b) Fold a $10 \times 10$ square from a $1 \times (100+9\sqrt3)$ rectangular strip (approximately $1\times 115.58$).
The strip can be bent, but not torn.

2006 Sharygin Geometry Olympiad First Round p6 grades 8-9 (a), 9-10 (b)
a) Given a segment $AB$ with a point $C$ inside it, which is the chord of a circle of radius $R$.
Inscribe in the formed segment a circle tangent to point $C$ and to the circle of radius $R$.
b) Given a segment $AB$ with a point $C$ inside it, which is the point of tangency of a circle of radius $r$. Draw through $A$ and $B$ a circle tangent to a circle of radius $r$.

2006 Sharygin Geometry Olympiad First Round p7 grades 8-10
The point $E$ is taken inside the square $ABCD$, the point $F$ is taken outside, so that the triangles $ABE$ and $BCF$ are congruent . Find the angles of the triangle $ABE$, if it is known that$EF$ is equal to the side of the square, and the angle $BFD$ is right.

2006 Sharygin Geometry Olympiad First Round p8 grades 8-9
The segment $AB$ divides the square into two parts, in each of which a circle can be inscribed. The radii of these circles are equal to $r_1$ and $r_2$ respectively, where $r_1> r_2$. Find the length of $AB$.

2006 Sharygin Geometry Olympiad First Round p9 grades 8-10
$L(a)$ is the line connecting the points of the unit circle corresponding to the angles $a$ and $\pi - 2a$. Prove that if $a + b + c = 2\pi$, then the lines $L (a), L (b)$ and $L (c)$ intersect at one point.

2006 Sharygin Geometry Olympiad First Round p10 grades 8-11
At what $n$ can a regular $n$-gon be cut by disjoint diagonals into $n- 2$ isosceles (including equilateral) triangles?

2006 Sharygin Geometry Olympiad First Round p11 grades 9-10
In the triangle $ABC, O$ is the center of the circumscribed circle, $A ', B', C '$ are the symmetrics of $A, B, C$ with respect to opposite sides, $A_1, B_1, C_1$ are the intersection points of the lines $OA'$ and $BC, OB'$ and $AC, OC'$ and $AB$. Prove that the lines $A A_1, BB_1, CC_1$ intersect at one point.

2006 Sharygin Geometry Olympiad First Round p12 grades 9-10 
In the triangle $ABC$, the bisector of angle $A$ is equal to the half-sum of the height and median drawn from vertex $A$. Prove that if $\angle A$ is obtuse, then $AB = AC$.

2006 Sharygin Geometry Olympiad First Round p13 grades 9-10

Two straight lines $a$ and $b$ are given and also points $A$ and $B$. Point $X$ slides along the line $a$, and point $Y$ slides along the line $b$, so that $AX // BY$. Find the locus of the intersection point of $AY$ with $XB$.

2006 Sharygin Geometry Olympiad First Round p14 grades 9-11 

Given a circle and a fixed point $P$ not lying on it. Find the geometrical locus of the orthocenters of the triangles $ABP$, where $AB$ is the diameter of the circle.

2006 Sharygin Geometry Olympiad First Round p15 grades 9-11
A circle is circumscribed around triangle $ABC$ and a circle is inscribed in it, which touches the sides of the triangle BC,CA,AB at points $A_1,B_1,C_1$, respectively. The line $B_1C_1$ intersects the line $BC$ at the point $P$, and $M$ is the midpoint of the segment $PA_1$. Prove that the segments of the tangents drawn from the point $M$ to the inscribed and circumscribed circle are equal.

2006 Sharygin Geometry Olympiad First Round p16 grades 9-11 

Regular triangles are built on the sides of the triangle $ABC$. It turned out that their vertices form a regular triangle. Is the original triangle regular also?

2006 Sharygin Geometry Olympiad First Round p17 grades 9-11

In two circles intersecting at points $A$ and $B$, parallel chords $A_1B_1$ and $A_2B_2$ are drawn. The lines $AA_1$ and $BB_2$ intersect at the point $X, AA_2$ and $BB_1$ intersect at the point $Y$. Prove that $XY // A_1B_1$

2006 Sharygin Geometry Olympiad First Round p18 grades 9-11 

Two perpendicular lines are drawn through the orthocenter $H$ of triangle $ABC$, one of which intersects $BC$ at point $X$, and the other intersects $AC$ at point $Y$. Lines $AZ, BZ$ are parallel, respectively with $HX$ and $HY$. Prove that the points $X, Y, Z$ lie on the same line.

2006 Sharygin Geometry Olympiad First Round p19 grades 10-11 

Through the midpoints of the sides of the triangle $T$, straight lines are drawn perpendicular to the bisectors of the opposite angles of the triangle. These lines formed a triangle $T_1$. Prove that the center of the circle circumscribed about $T_1$ is in the midpoint of the segment formed by the center of the inscribed circle and the intersection point of the heights of triangle $T$.

2006 Sharygin Geometry Olympiad First Round p20 grades 10-11

Through the midpoints of the sides of the triangle $T$, straight lines are drawn perpendicular to the bisectors of the opposite angles of the triangle. These lines formed a triangle $T_1$. Prove that the center of the circle circumscribed about $T_1$ is in the midpoint of the segment formed by the center of the inscribed circle and the intersection point of the heights of triangle $T$.

2006 Sharygin Geometry Olympiad First Round p21 grades 10-11 

On the sides $AB, BC, CA$ of triangle $ABC$, points $C', A', B'$ are taken.
Prove that for the areas of the corresponding triangles, the inequality holds:
$$S_{ABC}S^2_{A'B'C'}\ge  4S_{AB'C'}S_{BC'A'}S_{CA'B'}$$ and equality is achieved if and only if the lines $AA', BB', CC'$ intersect at one point.

2006 Sharygin Geometry Olympiad First Round p22 grades 10-11 

Given points $A, B$ on a circle and a point $P$ not lying on the circle. $X$ is an arbitrary point of the circle, $Y$ is the intersection point of lines $AX$ and $BP$. Find the locus of the centers of the circles circumscribed around the triangles $PXY$.

2006 Sharygin Geometry Olympiad First Round p23 grades 9-10 (a), 10-11 (b) 

$ABCD$ is a convex quadrangle, $G$ is its center of gravity as a homogeneous plate (i.e., the intersection point of two lines, each of which connects the centroids of triangles having a common diagonal).
a) Suppose that around $ABCD$ we can circumscribe a circle centered on $O$. We define $H$ similarly to $G$, taking orthocenters instead of centroids. Then the points of $H, G, O$ lie on the same line and $HG: GO = 2: 1$.
b) Suppose that in $ABCD$ we can inscribe a circle centered on $I$. The Nagel point N of the circumscribed quadrangle is the intersection point of two lines, each of which passes through points on opposite sides of the quadrangle that are symmetric to the tangent points of the inscribed circle relative to the midpoints of the sides. (These lines divide the perimeter of the quadrangle in half). Then $N, G, I$ lie on one straight line, with $NG: GI = 2: 1$.

2006 Sharygin Geometry Olympiad First Round p24 grades 9-10 (a), 10-11 (b) 

a) Two perpendicular rays are drawn through a fixed point $P$ inside a given circle, intersecting the circle at points $A$ and $B$. Find the geometric locus of the projections of $P$ on the lines $AB$.
b) Three pairwise perpendicular rays passing through the fixed point $P$ inside a given sphere intersect the sphere at points $A, B, C$. Find the geometrical locus of the projections $P$ on the $ABC$ plane.

2006 Sharygin Geometry Olympiad First Round p25 grade 11

In the tetrahedron ABCD , the dihedral angles at the $BC, CD$, and $DA$ edges are equal to $\alpha$, and for the remaining edges equal to $\beta$. Find the ratio $AB / CD$.

2006 Sharygin Geometry Olympiad First Round p26 grade 11

Four cones are given with a common vertex and the same generatrix, but with, generally speaking, different radii of the bases. Each of them is tangent to two others. Prove that the four tangent points of the circles of the bases of the cones lie on the same circle.

2005-2006 First Round

                                                                                                                collected inside aops here

grade 8

2006 Sharygin Geometry Olympiad Finals 8.1

Inscribe the equilateral triangle of the largest perimeter in a given semicircle.

2006 Sharygin Geometry Olympiad Finals 8.2

What $n$ is the smallest such that “there is a $n$-gon that can be cut into a triangle, a quadrilateral, ..., a $2006$-gon''?

2006 Sharygin Geometry Olympiad Finals 83

A parallelogram $ABCD$ is given. Two circles with centers at the vertices $A$ and $C$ pass through $B$. The straight line $\ell$ that passes through $B$ and crosses the circles at second time at points $X, Y$ respectively. Prove that $DX = DY$.

2006 Sharygin Geometry Olympiad Finals 8.4

Two equal circles intersect at points $A$ and $B$. $P$ is the point of one of the circles that is different from $A$ and $B, X$ and $Y$ are the second intersection points of the lines of $PA, PB$ with the other circle. Prove that the line passing through $P$ and perpendicular to $AB$ divides one of the arcs $XY$ in half.

2006 Sharygin Geometry Olympiad Finals 8.5

Is there a convex polygon with each side equal to some diagonal, and each diagonal equal to some side?

2006 Sharygin Geometry Olympiad Finals 8.6

A triangle $ABC$ and a point $P$ inside it are given. $A', B', C'$ are the projections of $P$ onto the straight lines ot the sides $BC,CA,AB$. Prove that the center of the circle circumscribed around the triangle $A'B'C'$ lies inside the triangle $ABC$.

grade 9

2006 Sharygin Geometry Olympiad Finals 9.1

Given a circle of radius $K$. Two other circles, the sum of the radii of which are also equal to $K$, tangent to the circle from the inside. Prove that the line connecting the points of tangency passes through one of the common points of these circles.

2006 Sharygin Geometry Olympiad Finals 9.2

Given a circle, point $A$ on it and point $M$ inside it. We consider the chords $BC$ passing through $M$. Prove that the circles passing through the midpoints of the sides of all the triangles $ABC$ are tangent to a fixed circle.

2006 Sharygin Geometry Olympiad Finals 9.3

Triangles $ABC$ and $A_1B_1C_1$ are similar and differently oriented. On the segment $AA_1$, a point $A'$ is taken such that $AA' / A_1A'= BC / B_1C_1$. We similarly construct $B'$ and $C'$. Prove that $A', B',C'$ lie on one straight line.

2006 Sharygin Geometry Olympiad Finals 9.4

In a non-convex hexagon, each angle is either $90$ or $270$ degrees. Is it true that for some lengths of the sides it can be cut into two hexagons similar to it and unequal to each other?

2006 Sharygin Geometry Olympiad Finals 9.5

A straight line passing through the center of the circumscribed circle and the intersection point of the heights of the non-equilateral triangle $ABC$ divides its perimeter and area in the same ratio.Find this ratio.

2006 Sharygin Geometry Olympiad Finals 9.6

A convex quadrilateral $ABC$ is given. $A',B',C',D'$ are the orthocenters of triangles $BCD, CDA, DAB, ABC$ respectively. Prove that in the quadrilaterals $ABCP$ and $A'B'C'D'$, the corresponding diagonals share the intersection points in the same ratio.

grade 10

2006 Sharygin Geometry Olympiad Finals p10.1  (translation incomplete)

Five lines go through one point. Prove that there exists a closed five-segment polygonal line, the vertices and the middle of the segments of which lie on these lines, and each line has exactly one vertex.

2006 Sharygin Geometry Olympiad Finals 10.2
The projections of the point $X$ onto the sides of the $ABCD$ quadrangle lie on the same circle. $Y$ is a point symmetric to $X$ with respect to the center of this circle. Prove that the projections of the point $B$ onto the lines $AX,XC, CY, YA$ also lie on the same circle.

2006 Sharygin Geometry Olympiad Finals 10.3

Given a circle and a point $P$ inside it, different from the center. We consider pairs of circles tangent to the given internally and to each other at point $P$. Find the locus of the points of intersection of the common external tangents to these circles.

2006 Sharygin Geometry Olympiad Finals 10.4

Lines containing the medians of the triangle $ABC$ intersect its circumscribed circle for a second time at the points $A_1, B_1, C_1$. The straight lines passing through $A,B,C$ parallel to opposite sides intersect it at points $A_2, B_2, C_2$. Prove that lines $A_1A_2,B_1B_2,C_1C_2$ intersect at one point.

2006 Sharygin Geometry Olympiad Finals 10.5

Can a tetrahedron scan turn out to be a triangle with sides $3, 4$ and $5$ (a tetrahedron can be cut only along the edges)?

2006 Sharygin Geometry Olympiad Finals 10.6

A quadrangle was drawn on the board, that you can inscribe and circumscribe a circle. Marked are the centers of these circles and the intersection point of the lines connecting the midpoints of the opposite sides, after which the quadrangle itself was erased. Restore it with a compass and ruler.

2006-2007 First Round

2007 Sharygin Geometry Olympiad First Round p1 grade 8

A triangle is cut into several (not less than two) triangles. One of them is isosceles (not equilateral), and all others are equilateral. Determine the angles of the original triangle.

2007 Sharygin Geometry Olympiad First Round p2 grade 8 

Each diagonal of a quadrangle divides it into two isosceles triangles. Is it true that the quadrangle is a diamond?

2007 Sharygin Geometry Olympiad First Round p3 grades 8-9

Segments connecting an inner point of a convex non-equilateral n-gon to its vertices divide the n-gon into n equal triangles. What is the least possible n?

2007 Sharygin Geometry Olympiad First Round p4 grade 8

Does a parallelogram exist such that all pairwise meets of bisectors of its angles are situated outside it?

2007 Sharygin Geometry Olympiad First Round p5 grades 8 / 8-10

A non-convex $n$-gon is cut into three parts by a straight line, and two parts are put together so that the resulting polygon is equal to the third part. Can $n$ be equal to:
a) five?
b) four?

2007 Sharygin Geometry Olympiad First Round p6 grades 8-9 / 10-11

a) What can be the number of symmetry axes of a checked polygon, that is, of a polygon whose sides lie on lines of a list of checked paper? (Indicate all possible values.)
b) What can be the number of symmetry axes of a checked polyhedron, that is, of a polyhedron consisting of equal cubes which border one to another by plane facets?

2007 Sharygin Geometry Olympiad First Round p7 grades 8-9

A convex polygon is circumscribed around a circle. Points of contact of its sides with the circle form a polygon with the same set of angles (the order of angles may differ). Is it true that the polygon is regular?

2007 Sharygin Geometry Olympiad First Round p8 grades 8-9

Three circles pass through a point $P$, and the second points of their intersection $A, B, C$ lie on a straight line. Let $A_1 B_1, C_1$ be the second meets of lines $AP, BP, CP$ with the corresponding circles. Let $C_2$ be the meet of lines AB_1 and $BA_1$. $Let A_2, B_2$ be defined similarly. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are equal,

2007 Sharygin Geometry Olympiad First Round p9 grades 8-9

Suppose two convex quadrangles are such that the sides of each of them lie on the middle perpendiculars to the sides of the other one. Determine their angles,

2007 Sharygin Geometry Olympiad First Round p10 grades 8-9

Find the locus of centers of regular triangles such that three given points $A, B, C$ lie respectively on three lines containing sides of the triangle.

2007 Sharygin Geometry Olympiad First Round p11 grades 8-10

A boy and his father are standing on a seashore. If the boy stands on his tiptoes, his eyes are at a height of $1$ m above sea-level, and if he seats on father’s shoulders, they are at a height of $2$ m. What is the ratio of distances visible for him in two eases?
(Find the answer to $0,1$, assuming that the radius of Earth equals $6000$ km.)

2007 Sharygin Geometry Olympiad First Round p12 grades 9-10

A rectangle $ABCD$ and a point $P$ are given. Lines passing through $A$ and $B$ and perpendicular to $PC$ and $PD$ respectively, meet at a point $Q$. Prove that $PQ \perp AB$.

2007 Sharygin Geometry Olympiad First Round p13 grades 9-10

On the side $AB$ of a triangle $ABC$, two points $X, Y$ are chosen so that $AX = BY$. Lines $CX$ and $CY$ meet the circumcircle of the triangle, for the second time, at points $U$ and $V$. Prove that all lines $UV$ (for all $X, Y$, given $A, B, C$) have a common point.

2007 Sharygin Geometry Olympiad First Round p14 grades 9-11

In a trapezium with bases $AD$ and $BC$, let $P$ and $Q$ be the middles of diagonals $AC$ and $BD$ respectively. Prove that if $\angle DAQ = \angle CAB$ then $\angle PBA = \angle DBC$.

2007 Sharygin Geometry Olympiad First Round p15 grades 9-11

In a triangle $ABC$, let $AA', BB'$ and $CC'$ be the bisectors. Suppose $A'B' \cap CC' =P$ and $A'C' \cap BB'= Q$. Prove that $\angle PAC = \angle QAB$.

2007 Sharygin Geometry Olympiad First Round p16 grades 9-11

On two sides of an angle, points $A, B$ are chosen. The middle $M$ of the segment $AB$ belongs to two lines such that one of them meets the sides of the angle at points $A_1, B_1$, and the other at points $A_2, B_2$. The lines $A_1B_2$ and $A_2B_1$ meet $AB$ at points $P$ and $Q$. Prove that $M$ is the middle of $PQ$.

2007 Sharygin Geometry Olympiad First Round p17 grades 9-11

What triangles can be cut into three triangles having equal radii of circumcircles?

2007 Sharygin Geometry Olympiad First Round p18 grades 9-11

Determine the locus of vertices of triangles which have prescribed orthocenter and center of circumcircle.

2007 Sharygin Geometry Olympiad First Round p19 grades 10-11

Into an angle $A$ of size $a$, a circle is inscribed tangent to its sides at points $B$ and $C$. A line tangent to this circle at a point M meets the segments $AB$ and $AC$ at points $P$ and $Q$ respectively. What is the minimum $a$ such that the inequality $S_{PAQ}<S_{BMC}$ is possible?

2007 Sharygin Geometry Olympiad First Round p20 grade 11

The base of a pyramid is a regular triangle having side of size $1$. Two of three angles at the vertex of the pyramid are right. Find the maximum value of the volume of the pyramid.

2007 Sharygin Geometry Olympiad First Round p21 grade 11

There are two pipes on the plane (the pipes are circular cylinders of equal size, $4$ m around). Two of them are parallel and, being tangent one to another in the common generatrix, form a tunnel over the plane. The third pipe is perpendicular to two others and cuts out a chamber in the tunnel. Determine the area of ​​the surface of this chamber.

2006-2007 Final Round

grade 8

2007 Sharygin Geometry Olympiad Finals 8.1

Determine on which side is the steering wheel disposed in the car depicted in the figure.

2007 Sharygin Geometry Olympiad  Finals 8.2

By straightedge and compass, reconstruct a right triangle $ABC$ ($\angle C = 90 ^ o$), given the vertices $A, C$ and a point on the bisector of angle $B$.

2007 Sharygin Geometry Olympiad  Finals 8.3

The diagonals of a convex quadrilateral dissect it into four similar triangles. Prove that this quadrilateral can also be dissected into two congruent triangles.

2007 Sharygin Geometry Olympiad Finals 8.4

Determine the locus of orthocenters of triangles, given the midpoint of a side and the bases of the altitudes drawn to two other sides.

2007 Sharygin Geometry Olympiad Finals 8.5

Medians $AA'$ and $BB'$ of triangle $ABC$ meet at point $M$, and $\angle AMB = 120^o$. Prove that angles $AB'M$ and $BA'M$ are neither both acute nor both obtuse.

2007 Sharygin Geometry Olympiad Finals 8.6

Two non-congruent triangles are called analogous if they can be denoted as $ABC$ and $A'B'C'$ such that $AB = A'B', AC = A'C'$ and $\angle B = \angle B'$ . Do there exist three mutually analogous triangles?

grade 9

2007 Sharygin Geometry Olympiad Finals 9.1

Given a circumscribed quadrilateral $ABCD$. Prove that its inradius is smaller than the sum of the inradii of triangles $ABC$ and $ACD$.

2007 Sharygin Geometry Olympiad Finals 9.2

Points $E$ and $F$ are chosen on the base side $AD$ and the lateral side $AB$ of an isosceles trapezoid $ABCD$, respectively. Quadrilateral $CDEF$ is an isosceles trapezoid as well. Prove that $AE \cdot ED = AF \cdot FB$.

2007 Sharygin Geometry Olympiad Finals 9.3

Given a hexagon $ABCDEF$ such that $AB=BC$, $CD=DE$ , $EF=FA$ and $\angle A = \angle C = \angle E$ Prove that $AD, BE, CF$ are concurrent.

2007 Sharygin Geometry Olympiad Finals 9.4 

Given a triangle $ABC$. An arbitrary point $P$ is chosen on the circumcircle of triangle $ABH$ ($H$ is the orthocenter of triangle $ABC$). Lines $AP$ and $BP$ meet the opposite sidelines of the triangle at points $A'$ and $B'$, respectively. Determine the locus of midpoints of segments $A'B'$.

2007 Sharygin Geometry Olympiad Finals 9.5

Reconstruct a triangle, given the incenter, the midpoint of some side and the base of the altitude drawn to this side.

2007 Sharygin Geometry Olympiad Finals 9.6

A cube with edge length $2n+ 1$ is dissected into small cubes of size  $1\times 1\times 1$ and bars of size $2\times 2\times 1$. Find the least possible number of cubes in such a dissection.

grade 10

2007 Sharygin Geometry Olympiad Finals 10.1

In an acute triangle $ABC$, altitudes at vertices $A$ and $B$ and bisector line at angle $C$ intersect the circumcircle again at points $A_1, B_1$ and $C_0$. Using the straightedge and compass, reconstruct the triangle by points $A_1, B_1$ and $C_0$.

2007 Sharygin Geometry Olympiad Finals 10.2

Points $A', B', C'$ are the bases of the altitudes $AA', BB'$ and $CC'$ of an acute triangle $ABC$. A circle with center $B$ and radius $BB'$ meets line $A'C'$ at points $K$ and $L$ (points $K$ and $A$ are on the same side of line $BB'$). Prove that the intersection point of lines $AK$ and $CL$ belongs to line $BO$ ($O$ is the circumcenter of triangle $ABC$).

2007 Sharygin Geometry Olympiad Finals 10.3

Given two circles intersecting at points $P$ and $Q$. Let C be an arbitrary point distinct from $P$ and $Q$ on the former circle. Let lines $CP$ and $CQ$ intersect again the latter circle at points A and B, respectively. Determine the locus of the circumcenters of triangles $ABC$.

2007 Sharygin Geometry Olympiad Finals 10.4

A quadrilateral A$BCD$ is inscribed into a circle with center $O$. Points $C', D'$ are the reflections of the orthocenters of triangles $ABD$ and $ABC$ at point $O$. Lines $BD$ and $BD'$ are symmetric with respect to the bisector of angle $ABC$. Prove that lines $AC$ and $AC'$ are symmetric with respect to the bisector of angle $DAB$.

2007 Sharygin Geometry Olympiad Finals 10.5

Each edge of a convex polyhedron is shifted such that the obtained edges form the frame of another convex polyhedron. Are these two polyhedra necessarily congruent?

2007 Sharygin Geometry Olympiad Finals 10.6 

Given are two concentric circles $\Omega$ and $\omega$. Each of the circles $b_1$ and $b_2$ is externally tangent to $\omega$ and internally tangent to $\Omega$, and $\omega$ each of the circles $c_1$ and $c_2$ is internally tangent to both $\Omega$ and $\omega$. Mark each point where one of the circles $b_1, b_2$ intersects one of the circles $c_1$ and $c_2$. Prove that there exist two circles distinct from $b_1, b_2, c_1, c_2$ which contain all $8$ marked points. (Some of these new circles may appear to be lines.)

2007-2008 First Round

2008 Sharygin Geometry Olympiad First Round p1 grade 8

Does a regular polygon exist such that just half of its diagonals are parallel to its sides?

by B.Frenkin

2008 Sharygin Geometry Olympiad First Round p2 grade 8

For a given pair of circles, construct two concentric circles such that both are tangent to the given two. What is the number of solutions, depending on location of the circles?

by V.Protasov

2008 Sharygin Geometry Olympiad First Round p3 grade 8

A triangle can be dissected into three equal triangles. Prove that some its angle is equal to $60^{\circ}$.

by A.Zaslavsky

2008 Sharygin Geometry Olympiad First Round p4 grades 8-9

The bisectors of two angles in a cyclic quadrilateral are parallel. Prove that the sum of squares of some two sides in the quadrilateral equals the sum of squares of two remaining sides.

by D.Shnol

2008 Sharygin Geometry Olympiad First Round p5 grades 8-9

Reconstruct the square $ABCD$, given its vertex $A$ and distances of vertices $B$ and $D$ from a fixed point $O$ in the plane.

by Kiev olympiad

2008 Sharygin Geometry Olympiad First Round p6 grades 8-9

In the plane, given two concentric circles with the center $A$. Let $B$ be an arbitrary point on some of these circles, and $C$ on the other one. For every triangle $ABC$, consider two equal circles mutually tangent at the point $K$, such that one of these circles is tangent to the line $AB$ at point $B$ and the other one is tangent to the line $AC$ at point $C$. Determine the locus of points $K$.

by A. Myakishev

2008 Sharygin Geometry Olympiad First Round p7 grades 8-9

Given a circle and a point $O$ on it. Another circle with center $O$ meets the first one at points $P$ and $Q$. The point $C$ lies on the first circle, and the lines $CP$, $CQ$ meet the second circle for the second time at points $A$ and $B$. Prove that $AB=PQ$.

by A.Zaslavsky

2008 Sharygin Geometry Olympiad First Round p8 grades 8-11

a) Prove that for $n > 4$, any convex $n$-gon can be dissected into $n$ obtuse triangles.

b) Prove that for any $n$, there exists a convex $n$-gon which cannot be dissected into less than $n$ obtuse triangles.

c) In a dissection of a rectangle into obtuse triangles, what is the least possible number of triangles? 

by T.Golenishcheva-Kutuzova, B.Frenkin

2008 Sharygin Geometry Olympiad First Round p9 grades 9-10

The reflections of diagonal $BD$ of a quadrilateral $ABCD$ in the bisectors of angles $B$ and $D$ pass through the midpoint of diagonal $AC$. Prove that the reflections of diagonal $AC$ in the bisectors of angles $A$ and $C$ pass through the midpoint of diagonal $BD$

(There was an error in published condition of this problem).

by A.Zaslavsky

2008 Sharygin Geometry Olympiad First Round p10 grades 9-10

Quadrilateral $ABCD$ is circumscribed arounda circle with center $I$. Prove that the projections of points $B$ and $D$ to the lines $IA$ and $IC$ lie on a single circle.

by A.Zaslavsky

2008 Sharygin Geometry Olympiad First Round p11 grades 9-10

Given four points $A$, $B$, $C$, $D$. Any two circles such that one of them contains $A$ and $B$, and the other one contains $C$ and $D$, meet. Prove that common chords of all these pairs of circles pass through a fixed point.

by A.Zaslavsky

2008 Sharygin Geometry Olympiad First Round p12 grades 9-10

Given a triangle $ABC$. Point $A_1$ is chosen on the ray $BA$ so that segments $BA_1$ and $BC$ are equal. Point $A_2$ is chosen on the ray $CA$ so that segments $CA_2$ and $BC$ are equal. Points  $B_1$, $B_2$ and $C_1$, $C_2$ are chosen similarly. Prove that lines $A_1A_2$, $B_1B_2$, $C_1C_2$ are parallel.

by A.Myakishev

2008 Sharygin Geometry Olympiad First Round p13 grades 9-10

Given triangle $ABC$. One of its excircles is tangent to the side $BC$ at point $A_1$ and to the extensions of two other sides. Another excircle is tangent to side $AC$ at point $B_1$. Segments $AA_1$ and $BB_1$ meet at point $N$. Point $P$ is chosen on the ray $AA_1$ so that $AP= NA_1$. Prove that $P$ lies on the incircle.

by A.Myakishev

2008 Sharygin Geometry Olympiad First Round p14 grades 9-10

The Euler line of a non-isosceles triangle is parallel to the bisector of one of its angles. Determine this angle

 (There was an error in published condition of this problem).

by V.Protasov

2008 Sharygin Geometry Olympiad First Round p15 grades 9-11

Given two circles and point $P$ not lying on them. Draw a line through $P$ which cuts chords of equal length from these circles.

by M.Volchkevich

2008 Sharygin Geometry Olympiad First Round p16 grades 9-11

Given two circles. Their common external tangent is tangent to them at points $A$ and $B$. Points $X$, $Y$ on these circles are such that some circle is tangent to the given two circles at these points, and in similar way (external or internal). Determine the locus of intersections of lines $AX$ and $BY$.

by A.Zaslavsky

2008 Sharygin Geometry Olympiad First Round p17 grades 9-11

Given triangle $ABC$ and a ruler with two marked intervals equal to $AC$ and $BC$. By this ruler only, find the incenter of the triangle formed by medial lines of triangle $ABC$.

by A.Myakishev

2008 Sharygin Geometry Olympiad First Round p18 grades 9-11

Prove that the triangle having sides $a$, $b$, $c$ and area $S$ satisfies the inequality
$a^2 + b^2 + c^2 - \frac12(|a -b|+|b-c| + |c-a|)^2\geq 4\sqrt3 S.$

by A.Abdullayev 

2008 Sharygin Geometry Olympiad First Round p19 grades 10-11

Given parallelogram $ABCD$ such that $AB = a$, $AD = b$. The first circle has its center at vertex $A$ and passes through $D$, and the second circle has its center at $C$ and passes through $D$. A circle with center $B$ meets the first circle at points $M_1$, $N_1$, and the second circle at points $M_2$, $N_2$. Determine the ratio $M_1N_1/M_2N_2$.

by V.Protasov

2008 Sharygin Geometry Olympiad First Round p20 grades 10-11

a) Some polygon has the following property:
if a line passes through two points which bisect its perimeter then this line bisects the area of the polygon. Is it true that the polygon is central symmetric?
b) Is it true that any figure with the property from part a) is central symmetric?

by A.Zaslavsky

2008 Sharygin Geometry Olympiad First Round p21 grades 10-11

In a triangle, one has drawn perpendicular bisectors to its sides and has measured their segments lying inside the triangle.
a) All three segments are equal. Is it true that the triangle is equilateral?
b) Two segments are equal. Is it true that the triangle is isosceles?
c) Can the segments have length 4, 4 and 3?

by A.Zaslavsky, B.Frenkin

2008 Sharygin Geometry Olympiad First Round p22 grades 10-11

a) All vertices of a pyramid lie on the facets of a cube but not on its edges, and each facet contains at least one vertex. What is the maximum possible number of the vertices of the pyramid?
b) All vertices of a pyramid lie in the facet planes of a cube but not on the lines including its edges, and each facet plane contains at least one vertex. What is the maximum possible number of the vertices of the pyramid?

by A.Khachaturyan

2008 Sharygin Geometry Olympiad First Round p23 grades 10-11

In the space, given two intersecting spheres of different radii and a point $A$ belonging to both spheres. Prove that there is a point $B$ in the space with the following property:
if an arbitrary circle passes through points $A$ and $B$ then the second points of its meet with the given spheres are equidistant from $B$.

by V.Protasov

2008 Sharygin Geometry Olympiad First Round p24 grade  11

Let $h$ be the least altitude of a tetrahedron, and $d$ the least distance between its opposite edges. For what values of $t$ the inequality $d>th$ is possible? 

by I.Bogdanov

2007-2008  Final Round

grade 8

2008 Sharygin Geometry Olympiad Finals 8.1

Does a convex quadrilateral without parallel sidelines exist such that it can be divided into four equal triangles?

by B.Frenkin 

2008 Sharygin Geometry Olympiad Finals 8.2

Given right triangle $ABC$ with hypothenuse $AC$ and $\angle A =50^{\circ}$. Points $K$ and $L$ on the cathetus $BC$ are such that $\angle KAC =  \angle LAB = 10^{\circ}$. Determine the ratio $CK/LB$.

by F.Nilov

2008 Sharygin Geometry Olympiad Finals 8.3
Two opposite angles of a convex quadrilateral with perpendicular diagonals are equal. Prove that a circle can be inscribed in this quadrilateral.

by D.Shnol

2008 Sharygin Geometry Olympiad Finals 8.4
Let $CC_0$ be a median of triangle $ABC$; the perpendicular bisectors to $AC$ and $BC$ intersect $CC_0$ in points $A'$, $B'$; $C_1$ is the meet of lines $AA'$ and $BB'$. Prove that $\angle C_1CA= \angle C_0CB$.

by F.Nilov, A.Zaslavsky

2008 Sharygin Geometry Olympiad Finals 8.5
Given two triangles $ABC$, $A'B'C'$. Denote by $\alpha$ the angle between the altitude and the median from vertex $A$ of triangle $ABC$. Angles $\beta$, $\gamma$, $\alpha'$, $\beta'$, $\gamma'$ are defined similarly. It is known that $\alpha = \alpha'$, $\beta =\beta'$, $\gamma =  \gamma'$. Can we conclude that the triangles are similar?

by A.Zaslavsky

2008 Sharygin Geometry Olympiad Finals 8.6
Consider the triangles such that all their vertices are vertices of a given regular $2008$-gon. What triangles are more numerous among them: acute-angled or obtuse-angled?

by B.Frenkin

2008 Sharygin Geometry Olympiad Finals 8.7
Given isosceles triangle $ABC$ with base $AC$ and $\angle B =\alpha$. The arc $AC$ constructed outside the triangle has angular measure equal to $\beta$. Two lines passing through $B$ divide the segment and the arc $AC$ into three equal parts. Find the ratio $\alpha / \beta$.

by F.Nilov

2008 Sharygin Geometry Olympiad Finals 8.8
A convex quadrilateral was drawn on the blackboard. Boris marked the centers of four excircles each touching one side of the quadrilateral and the extensions of two adjacent sides. After this, Alexey erased the quadrilateral. Can Boris define its perimeter?

by B.Frenkin, A.Zaslavsky

grade 9

2008 Sharygin Geometry Olympiad Finals 9/1

A convex polygon can be divided into $2008$ congruent quadrilaterals. Is it true that this polygon has a center or an axis of symmetry?

by A.Zaslavsky

2008 Sharygin Geometry Olympiad Finals 9.2
Given quadrilateral $ABCD$. Find the locus of points such that their projections to the lines $AB$, $BC$, $CD$, $DA$ form a quadrilateral with perpendicular diagonals.

by F.Nilov

2008 Sharygin Geometry Olympiad Finals 9.3
Prove the inequality $\frac1{\sqrt {2\sin A}} + \frac1{\sqrt {2\sin B}} +\frac1{\sqrt {2\sin C}}\leq\sqrt {\frac {p}{r}},$ where $p$ and $r$ are the semiperimeter and the inradius of triangle $ABC$.

by R.Pirkuliev

2008 Sharygin Geometry Olympiad Finals 9.4
Let $CC_0$ be a median of triangle $ABC$; the perpendicular bisectors to $AC$ and $BC$ intersect $CC_0$ in points $A_c$, $B_c$; $C_1$ is the common point of $AA_c$ and $BB_c$. Points $A_1$, $B_1$ are defined similarly. Prove that circle $A_1B_1C_1$ passes through the circumcenter of triangle $ABC$.

by F.Nilov, A.Zaslavsky

2008 Sharygin Geometry Olympiad Finals 9.5
Can the surface of a regular tetrahedron be glued over with equal regular hexagons?

by N.Avilov

2008 Sharygin Geometry Olympiad Finals 9.6
Construct the triangle, given its centroid and the feet of an altitude and a bisector from the same vertex.

by B.Frenkin

2008 Sharygin Geometry Olympiad Finals 9.7

The circumradius of triangle $ABC$ is equal to $R$. Another circle with the same radius passes through the orthocenter $H$ of this triangle and intersect its circumcirle in points $X$, $Y$. Point $Z$ is the fourth vertex of parallelogram $CXZY$. Find the circumradius of triangle $ABZ$.

by A.Zaslavsky

2008 Sharygin Geometry Olympiad Finals 9.8

Points $P$, $Q$ lie on the circumcircle $\omega$ of triangle $ABC$. The perpendicular bisector $l$ to $PQ$ intersects $BC$, $CA$, $AB$ in points $A'$, $B'$, $C'$. Let $A"$, $B"$, $C"$ be the second common points of $l$ with the circles $A'PQ$, $B'PQ$, $C'PQ$. Prove that $AA"$, $BB"$, $CC"$ concur.

by J.-L.Ayme, France

grade 10

2008 Sharygin Geometry Olympiad Finals 10.1

An inscribed and circumscribed $n$-gon is divided by some line into two inscribed and circumscribed polygons with different numbers of sides. Find $n$.

by B.Frenkin

2008 Sharygin Geometry Olympiad Finals 10.2

Let triangle $A_1B_1C_1$ be symmetric to $ABC$ wrt the incenter of its medial triangle. Prove that the orthocenter of $A_1B_1C_1$ coincides with the circumcenter of the triangle formed by the excenters of $ABC$.

by A.Myakishev

2008 Sharygin Geometry Olympiad Finals 10.3

Suppose $X$ and $Y$ are the common points of two circles $\omega_1$ and $\omega_2$. The third circle $\omega$ is internally tangent to $\omega_1$ and $\omega_2$ in $P$ and $Q$ respectively. Segment $XY$ intersects $\omega$ in points $M$ and $N$. Rays $PM$ and $PN$ intersect $\omega_1$ in points $A$ and $D$; rays $QM$ and $QN$ intersect $\omega_2$ in points $B$ and $C$ respectively. Prove that $AB = CD$.

by V.Yasinsky, Ukraine

2008 Sharygin Geometry Olympiad Finals 10.4

Given three points $C_0$, $C_1$, $C_2$ on the line $l$. Find the locus of incenters of triangles $ABC$ such that points $A$, $B$ lie on $l$ and the feet of the median, the bisector and the altitude from $C$ coincide with $C_0$, $C_1$, $C_2$.

by A.Zaslavsky

2008 Sharygin Geometry Olympiad Finals 10.5

A section of a regular tetragonal pyramid is a regular pentagon. Find the ratio of its side to the side of the base of the pyramid.

by I.Bogdanov

2008 Sharygin Geometry Olympiad Finals 10.6

The product of two sides in a triangle is equal to $8Rr$, where $R$ and $r$ are the circumradius and the inradius of the triangle. Prove that the angle between these sides is less than $60^{\circ}$.

by B.Frenkin

2008 Sharygin Geometry Olympiad Finals 10.7

Two arcs with equal angular measure are constructed on the medians $AA'$ and $BB'$ of triangle $ABC$ towards vertex $C$. Prove that the common chord of the respective circles passes through $C$.

by F.Nilov

2008 Sharygin Geometry Olympiad Finals 10.8

Given a set of points inn the plane. It is known that among any three of its points there are two such that the distance between them doesn't exceed 1. Prove that this set can be divided into three parts such that the diameter of each part does not exceed 1.

by A.Akopyan, V.Dolnikov

2008-2009 First Round

2009 Sharygin Geometry Olympiad First Round p1 grade  8

Points $B_1$ and $B_2$ lie on ray $AM$, and points $C_1$ and $C_2$ lie on ray $AK$. The circle with center $O$ is inscribed into triangles $AB_1C_1$ and $AB_2C_2$. Prove that the angles $B_1OB_2$ and $C_1OC_2$ are equal.

2009 Sharygin Geometry Olympiad First Round p2 grade  8

Given nonisosceles triangle $ABC$. Consider three segments passing through different vertices of this triangle and bisecting its perimeter. Are the lengths of these segments certainly different?

2009 Sharygin Geometry Olympiad First Round p3 grade  8

The bisectors of trapezoid's angles form a quadrilateral with perpendicular diagonals. Prove that this trapezoid is isosceles.

2009 Sharygin Geometry Olympiad First Round p4 grades  8-9

Let $P$ and $Q$ be the common points of two circles. The ray with origin $Q$ reflects from the first circle in points $A_1$, $A_2$,$\ldots$ according to the rule ''the angle of incidence is equal to the angle of reflection''. Another ray with origin $Q$ reflects from the second circle in the points $B_1$, $B_2$,$\ldots$ in the same manner. Points $A_1$, $B_1$ and $P$ occurred to be collinear. Prove that all lines $A_iB_i$ pass through P.

2009 Sharygin Geometry Olympiad First Round p5 grades  8-9

Given triangle $ABC$. Point $O$ is the center of the excircle touching the side $BC$. Point $O_1$ is the reflection of $O$ in $BC$. Determine angle $A$ if $O_1$ lies on the circumcircle of $ABC$.

2009 Sharygin Geometry Olympiad First Round p6 grades  8-9

Find the locus of excenters of right triangles with given hypotenuse.

2009 Sharygin Geometry Olympiad First Round p7 grades  8-9

Given triangle $ABC$. Points $M$, $N$ are the projections of $B$ and $C$ to the bisectors of angles $C$ and $B$ respectively. Prove that line $MN$ intersects sides $AC$ and $AB$ in their points of contact with the incircle of $ABC$.

2009 Sharygin Geometry Olympiad First Round p8 grades  8-10

Some polygon can be divided into two equal parts by three different ways. Is it certainly valid that this polygon has an axis or a center of symmetry?

2009 Sharygin Geometry Olympiad First Round p9 grade  9

Given $n$ points on the plane, which are the vertices of a convex polygon, $n > 3$. There exists $k$ regular triangles with the side equal to $1$ and the vertices at the given points.

• Prove that $k < \frac {2}{3}n$.
• Construct the configuration with $k > 0.666n$.

2009 Sharygin Geometry Olympiad First Round p10 grade 9

Let $ABC$ be an acute triangle, $CC_1$ its bisector, $O$ its circumcenter. The perpendicular from $C$ to $AB$ meets line $OC_1$ in a point lying on the circumcircle of $AOB$. Determine angle $C$.

2009 Sharygin Geometry Olympiad First Round p11 grade 9

Given quadrilateral $ABCD$. The circumcircle of $ABC$ is tangent to side $CD$, and the circumcircle of $ACD$ is tangent to side $AB$. Prove that the length of diagonal $AC$ is less than the distance between the midpoints of $AB$ and $CD$. 

2009 Sharygin Geometry Olympiad First Round p12 grades  9-10

Let $CL$ be a bisector of triangle $ABC$. Points $A_1$ and $B_1$ are the reflections of $A$ and $B$ in $CL$, points $A_2$ and $B_2$ are the reflections of $A$ and $B$ in $L$. Let $O_1$ and $O_2$ be the circumcenters of triangles $AB_1B_2$ and $BA_1A_2$ respectively. Prove that angles $O_1CA$ and $O_2CB$ are equal.

2009 Sharygin Geometry Olympiad First Round p13 grades  9-10

In triangle $ABC$, one has marked the incenter, the foot of altitude from vertex $C$ and the center of the excircle tangent to side $AB$. After this, the triangle was erased. Restore it.

2009 Sharygin Geometry Olympiad First Round p14 grades  9-10

Given triangle $ABC$ of area 1. Let $BM$ be the perpendicular from $B$ to the bisector of angle $C$. Determine the area of triangle $AMC$.

2009 Sharygin Geometry Olympiad First Round p15 grades  9-10

Given a circle and a point $C$ not lying on this circle. Consider all triangles $ABC$ such that points $A$ and $B$ lie on the given circle. Prove that the triangle of maximal area is isosceles.

2009 Sharygin Geometry Olympiad First Round p16 grades  9-11

Three lines passing through point $O$ form equal angles by pairs. Points $A_1$, $A_2$ on the first line and $B_1$, $B_2$ on the second line are such that the common point $C_1$ of $A_1B_1$ and $A_2B_2$ lies on the third line. Let $C_2$ be the common point of $A_1B_2$ and $A_2B_1$. Prove that angle $C_1OC_2$ is right.

2009 Sharygin Geometry Olympiad First Round p17 grades  9-11

Given triangle $ABC$ and two points $X$, $Y$ not lying on its circumcircle. Let $A_1$, $B_1$, $C_1$ be the projections of $X$ to $BC$, $CA$, $AB$, and $A_2$, $B_2$, $C_2$ be the projections of $Y$. Prove that the perpendiculars from $A_1$, $B_1$, $C_1$ to $B_2C_2$, $C_2A_2$, $A_2B_2$, respectively, concur if and only if line $XY$ passes through the circumcenter of $ABC$.

2009 Sharygin Geometry Olympiad First Round p18 grades  9-11

Given three parallel lines on the plane. Find the locus of incenters of triangles with vertices lying on these lines (a single vertex on each line).

2009 Sharygin Geometry Olympiad First Round p19 grades  10-11

Given convex $n$-gon $A_1\ldots A_n$. Let $P_i$ ($i =1,\ldots , n$) be such points on its boundary that $A_iP_i$ bisects the area of polygon. All points $P_i$ don't coincide with any vertex and lie on $k$ sides of $n$-gon. What is the maximal and the minimal value of $k$ for each given $n$?

2009 Sharygin Geometry Olympiad First Round p20 grades  10-11

Suppose $H$ and $O$ are the orthocenter and the circumcenter of acute triangle $ABC$; $AA_1$, $BB_1$ and $CC_1$ are the altitudes of the triangle. Point $C_2$ is the reflection of $C$ in $A_1B_1$. Prove that $H$, $O$, $C_1$ and $C_2$ are concyclic.

2009 Sharygin Geometry Olympiad First Round p21 grades  10-11

The opposite sidelines of quadrilateral $ABCD$ intersect at points $P$ and $Q$. Two lines passing through these points meet the side of $ABCD$ in four points which are the vertices of a parallelogram. Prove that the center of this parallelogram lies on the line passing through the midpoints of diagonals of $ABCD$.

2009 Sharygin Geometry Olympiad First Round p22 grades  10-11

Construct a quadrilateral which is inscribed and circumscribed, given the radii of the respective circles and the angle between the diagonals of quadrilateral.

2009 Sharygin Geometry Olympiad First Round p23 grades  10-11

Is it true that for each $n$, the regular $2n$-gon is a projection of some polyhedron having not greater than $n + 2$ faces?

2009 Sharygin Geometry Olympiad First Round p24 grade  11

A sphere is inscribed into a quadrangular pyramid. The point of contact of the sphere with the base of the pyramid is projected to the edges of the base. Prove that these projections are concyclic. 

2008-2009  Final Round

grade 8

2009 Sharygin Geometry Olympiad Finals 8.1

Minor base $BC$ of trapezoid $ABCD$ is equal to side $AB$, and diagonal $AC$ is equal to base $AD$. The line passing through B and parallel to $AC$ intersects line $DC$ in point $M$. Prove that $AM$ is the bisector of angle $\angle BAC$.

by A.Blinkov, Y.Blinkov

2009 Sharygin Geometry Olympiad Finals 8.2

A cyclic quadrilateral is divided into four quadrilaterals by two lines passing through its inner point. Three of these quadrilaterals are cyclic with equal circumradii. Prove that the fourth part also is cyclic quadrilateral and its circumradius is the same.

by A.Blinkov

2009 Sharygin Geometry Olympiad Finals 8.3

Let $AH_a$ and $BH_b$ be the altitudes of triangle $ABC$. Points $P$ and $Q$ are the projections of $H_a$ to $AB$ and $AC$. Prove that line $PQ$ bisects segment $H_aH_b$.

by A.Akopjan, K.Savenkov

2009 Sharygin Geometry Olympiad Finals 8.4

Given is $\triangle ABC$ such that $\angle A = 57^o, \angle B = 61^o$ and $\angle C = 62^o$. Which segment is longer: the angle bisector through $A$ or the median through $B$?

by N.Beluhov

2009 Sharygin Geometry Olympiad Finals 8.p5

Given triangle $ABC$. Point $M$ is the projection of vertex $B$ to bisector of angle $C$. $K$ is the touching point of the incircle with side $BC$. Find angle $\angle  MKB$ if $\angle BAC = \alpha$

by V.Protasov

2009 Sharygin Geometry Olympiad Finals 8.6

Can four equal polygons be placed on the plane in such a way that any two of them don't have common interior points, but have a common boundary segment?

by S.Markelov

2009 Sharygin Geometry Olympiad Finals 8.7

Let $s$ be the circumcircle of triangle $ABC, L$ and $W$ be common points of angle's $A$ bisector with side $BC$ and $s$ respectively, $O$ be the circumcenter of triangle $ACL$. Restore triangle $ABC$, if circle $s$ and points $W$ and $O$ are given.

by D.Prokopenko

2009 Sharygin Geometry Olympiad Finals 8.8

A triangle $ABC$ is given, in which the segment $BC$ touches the incircle and the corresponding excircle in points $M$ and $N$. If $\angle BAC = 2 \angle MAN$, show that $BC = 2MN$.

by N.Beluhov

grade 9

2009 Sharygin Geometry Olympiad Finals 9.1

The midpoint of triangle's side and the base of the altitude to this side are symmetric wrt the touching point of this side with the incircle. Prove that this side equals one third of triangle's perimeter.

by A.Blinkov, Y.Blinkov

2009 Sharygin Geometry Olympiad Finals 9.2

Given a convex quadrilateral $ABCD$. Let $R_a, R_b, R_c$ and $R_d$ be the circumradii of triangles $DAB, ABC, BCD, CDA$. Prove that inequality $R_a < R_b < R_c < R_d$ is equivalent to $180^o -  \angle CDB < \angle CAB < \angle CDB$ .

by O.Musin

2009 Sharygin Geometry Olympiad Finals 9.3

Quadrilateral $ABCD$ is circumscribed, rays $BA$ and $CD$ intersect in point $E$, rays $BC$ and $AD$ intersect in point $F$. The incircle of the triangle formed by lines $AB, CD$ and the bisector of angle $B$, touches $AB$ in point $K$, and the incircle of the triangle formed by lines $AD, BC$ and the bisector of angle $B$, touches $BC$ in point $L$. Prove that lines $KL, AC$ and $EF$ concur.

by I.Bogdanov

2009 Sharygin Geometry Olympiad Finals 9.4

Given regular $17$-gon $A_1 ... A_{17}$. Prove that two triangles formed by lines $A_1A_4, A_2A_{10}, A_{13}A_{14}$ and $A_2A_3, A_4A_6 A_{14}A_{15}$ are equal.

by N.Beluhov 

2009 Sharygin Geometry Olympiad Finals 9.5

Let $n$ points lie on the circle. Exactly half of triangles formed by these points are acute-angled. Find all possible $n$.

by B.Frenkin

2009 Sharygin Geometry Olympiad Finals 9.6

Given triangle $ABC$ such that $AB- BC = \frac{AC}{\sqrt2}$ . Let $M$ be the midpoint of $AC$, and $N$ be the base of the bisector from $B$. Prove that $\angle BMC + \angle BNC = 90^o$.

by A.Akopjan

2009 Sharygin Geometry Olympiad Finals 9.7

Given two intersecting circles with centers $O_1, O_2$. Construct the circle touching one of them externally and the second one internally such that the distance from its center to $O_1O_2$ is maximal.

by M.Volchkevich 

2009 Sharygin Geometry Olympiad Finals 9.8

Given cyclic quadrilateral $ABCD$. Four circles each touching its diagonals and the circumcircle internally are equal. Is $ABCD$ a square?

by C.Pohoata, A.Zaslavsky

grade 10

2009 Sharygin Geometry Olympiad Finals 10.1

Let $a, b, c$ be the lengths of some triangle's sides, $p, r$ be the semiperimeter and the inradius of triangle. Prove an inequality $\sqrt{\frac{ab(p- c)}{p}} +\sqrt{\frac{ca(p- b)}{p}} +\sqrt{\frac{bc(p-a)}{p}}   \ge 6r$

by D.Shvetsov 

2009 Sharygin Geometry Olympiad Finals 10.2 

Given quadrilateral $ABCD$. Its sidelines$AB$ and $CD$ intersect in point $K$. It's diagonals intersect in point $L$. It is known that line $KL$ pass through the centroid of $ABCD$. Prove that $ABCD$ is trapezoid.

by  F.Nilov

2009 Sharygin Geometry Olympiad Finals 10.3

The cirumradius and the inradius of triangle $ABC$ are equal to $R$ and $r, O, I$ are the centers of respective circles. External bisector of angle $C$ intersect $AB$ in point $P$. Point $Q$ is the projection of $P$ to line $OI$. Find distance $OQ.$

by A.Zaslavsky, A.Akopjan

2009 Sharygin Geometry Olympiad Finals 10.4

Three parallel lines $d_a, d_b, d_c$ pass through the vertex of triangle $ABC$. The reflections of $d_a, d_b, d_c$ in $BC, CA, AB$ respectively form triangle $XYZ$. Find the locus of incenters of such triangles.

by C.Pohoata

2009 Sharygin Geometry Olympiad Finals 10.5

Rhombus $CKLN$ is inscribed into triangle $ABC$ in such way that point $L$ lies on side $AB$, point $N$ lies on side $AC$, point $K$ lies on side $BC$. $O_1, O_2$ and $O$ are the circumcenters of triangles $ACL, BCL$ and $ABC$ respectively. Let $P$ be the common point of circles $ANL$ and $BKL$, distinct from $L$. Prove that points $O_1, O_2, O$ and $P$ are concyclic.

by D.Prokopenko

2009 Sharygin Geometry Olympiad Finals 10.6

Let $M, I$ be the centroid and the incenter of triangle $ABC, A_1$ and $B_1$ be the touching points of the incircle with sides $BC$ and $AC, G$ be the common point of lines $AA_1$ and $BB_1$. Prove that angle $\angle CGI$ is right if and only if $GM // AB$.

by A.Zaslavsky

2009 Sharygin Geometry Olympiad Finals 10.7

Given points $O, A_1, A_2, ..., A_n$  on the plane. For any two of these points the square of distance between them is natural number. Prove that there exist two vectors $\vec{x}$ and $\vec{y}$, such that for any point $A_i$, $\vec{OA_i }= k\vec{x}+l \vec{y}$, where $k$ and $l$ are some integer numbers.

by A.Glazyrin

2009 Sharygin Geometry Olympiad Finals 10.8

Can the regular octahedron be inscribed into regular dodecahedron in such way that all vertices of octahedron be the vertices of dodecahedron?

by B.Frenkin

2009-2010 First Round

2010 Sharygin Geometry Olympiad First Round p1 grade 8

Does there exist a triangle, whose side is equal to some of its altitudes, another side is equal to some of its bisectors, and the third is equal to some of its medians?

2010 Sharygin Geometry Olympiad First Round p2 grade 8

Bisectors $AA_1$ and $BB_1$ of a right triangle $ABC \ (\angle C=90^\circ )$ meet at a point $I.$ Let $O$ be the circumcenter of triangle $CA_1B_1.$ Prove that $OI \perp AB.$

2010 Sharygin Geometry Olympiad First Round p3 grade 8

Points $A', B', C'$ lie on sides $BC, CA, AB$ of triangle $ABC.$ for a point $X$ one has $\angle AXB =\angle A'C'B' + \angle ACB$ and $\angle BXC = \angle B'A'C' +\angle BAC.$ Prove that the quadrilateral $XA'BC'$ is cyclic.

2010 Sharygin Geometry Olympiad First Round p4 grade 8

The diagonals of a cyclic quadrilateral $ABCD$ meet in a point $N.$ The circumcircles of triangles $ANB$ and $CND$ intersect the sidelines $BC$ and $AD$ for the second time in points $A_1,B_1,C_1,D_1.$ Prove that the quadrilateral $A_1B_1C_1D_1$ is inscribed in a circle centered at $N.$

2010 Sharygin Geometry Olympiad First Round p5 grades 8-9

A point $E$ lies on the altitude $BD$ of triangle $ABC$, and $\angle AEC=90^\circ.$ Points $O_1$ and $O_2$ are the circumcenters of triangles $AEB$ and $CEB$; points $F, L$ are the midpoints of the segments $AC$ and $O_1O_2.$ Prove that the points $L,E,F$ are collinear.

2010 Sharygin Geometry Olympiad First Round p6 grades 8-9

Points $M$ and $N$ lie on the side $BC$ of the regular triangle $ABC$ ($M$ is between $B$ and $N$), and $\angle MAN=30^\circ.$ The circumcircles of triangles $AMC$ and $ANB$ meet at a point $K.$ Prove that the line $AK$ passes through the circumcenter of triangle $AMN.$

2010 Sharygin Geometry Olympiad First Round p7 grades 8-9

The line passing through the vertex $B$ of a triangle $ABC$ and perpendicular to its median $BM$ intersects the altitudes dropped from $A$ and $C$ (or their extensions) in points $K$ and $N.$ Points $O_1$ and $O_2$ are the circumcenters of the triangles $ABK$ and $CBN$ respectively. Prove that $O_1M=O_2M.$

2010 Sharygin Geometry Olympiad First Round p8 grades 8-10

Let $AH$ be the altitude of a given triangle $ABC.$ The points $I_b$ and $I_c$ are the incenters of the triangles $ABH$ and $ACH$ respectively. $BC$ touches the incircle of the triangle $ABC$ at a point $L.$ Find $\angle LI_bI_c.$

2010 Sharygin Geometry Olympiad First Round p9 grades 8-10

A point inside a triangle is called good  if three cevians passing through it are equal. Assume for an isosceles triangle $ABC \ (AB=BC)$ the total number of  good  points is odd. Find all possible values of this number.

2010 Sharygin Geometry Olympiad First Round p10 grades 8-11

Let three lines forming a triangle $ABC$ be given. Using a two-sided ruler and drawing at most eight lines construct a point $D$ on the side $AB$ such that $\frac{AD}{BD}=\frac{BC}{AC}.$

2010 Sharygin Geometry Olympiad First Round p11 grades 8-11

A convex $n-$gon is split into three convex polygons. One of them has $n$ sides, the second one has more than $n$ sides, the third one has less than $n$ sides. Find all possible values of $n.$

2010 Sharygin Geometry Olympiad First Round p12 grade 9

Let $AC$ be the greatest leg of a right triangle $ABC,$ and $CH$ be the altitude to its hypotenuse. The circle of radius $CH$ centered at $H$ intersects $AC$ in point $M.$ Let a point $B'$ be the reflection of $B$ with respect to the point $H.$ The perpendicular to $AB$ erected at $B'$ meets the circle in a point $K$. Prove that
a) $B'M \parallel BC$
b) $AK$ is tangent to the circle.

2010 Sharygin Geometry Olympiad First Round p13 grade 9

Let us have a convex quadrilateral $ABCD$ such that $AB=BC.$ A point $K$ lies on the diagonal $BD,$ and $\angle AKB+\angle  BKC=\angle  A + \angle  C.$ Prove that $AK \cdot CD = KC \cdot AD.$

2010 Sharygin Geometry Olympiad First Round p14 grades 9-10

We have a convex quadrilateral $ABCD$ and a point $M$ on its side $AD$ such that $CM$ and $BM$ are parallel to $AB$ and $CD$ respectively. Prove that $S_{ABCD} \geq 3 S_{BCM}.$


Remark. $S$ denotes the area function.

2010 Sharygin Geometry Olympiad First Round p15 grades 9-11

Let $AA_1, BB_1$ and $CC_1$ be the altitudes of an acute-angled triangle $ABC.$ $AA_1$ meets $B_1C_1$ in a point $K.$ The circumcircles of triangles $A_1KC_1$ and $A_1KB_1$ intersect the lines $AB$ and $AC$ for the second time at points $N$ and $L$ respectively. Prove that
a)  The sum of diameters of these two circles is equal to $BC,$
b) $\frac{A_1N}{BB_1} + \frac{A_1L}{CC_1}=1.$

2010 Sharygin Geometry Olympiad First Round p16 grades 9-11

A circle touches the sides of an angle with vertex $A$ at points $B$ and $C.$ A line passing through $A$ intersects this circle in points $D$ and $E.$ A chord $BX$ is parallel to $DE.$ Prove that $XC$ passes through the midpoint of the segment $DE.$

2010 Sharygin Geometry Olympiad First Round p17 grades 9-11

Construct a triangle, if the lengths of the bisectrix and of the altitude from one vertex, and of the median from another vertex are given.

2010 Sharygin Geometry Olympiad First Round p18 grades 9-11

A point $B$ lies on a chord $AC$ of circle $\omega.$ Segments $AB$ and $BC$ are diameters of circles $\omega_1$ and $\omega_2$ centered at $O_1$ and $O_2$ respectively. These circles intersect $\omega$ for the second time in points $D$ and $E$ respectively. The rays $O_1D$ and $O_2E$ meet in a point $F,$ and the rays $AD$ and $CE$ do in a point $G.$ Prove that the line $FG$ passes through the midpoint of the segment $AC.$

2010 Sharygin Geometry Olympiad First Round p19 grades 9-11

A quadrilateral $ABCD$ is inscribed into a circle with center $O.$ Points $P$ and $Q$ are opposite to $C$ and $D$ respectively. Two tangents drawn to that circle at these points meet the line $AB$ in points $E$ and $F.$ ($A$ is between $E$ and $B$, $B$ is between $A$ and $F$). The line $EO$ meets $AC$ and $BC$ in points $X$ and $Y$ respectively, and the line $FO$ meets $AD$ and $BD$ in points $U$ and $V$ respectively. Prove that $XV=YU.$

2010 Sharygin Geometry Olympiad First Round p20 grade 10

The incircle of an acute-angled triangle $ABC$ touches $AB, BC, CA$ at points $C_1, A_1, B_1$ respectively. Points $A_2, B_2$ are the midpoints of the segments $B_1C_1, A_1C_1$ respectively. Let $P$ be a common point of the incircle and the line $CO$, where $O$ is the circumcenter of triangle $ABC.$ Let also $A'$ and $B'$ be the second common points of $PA_2$ and $PB_2$ with the incircle. Prove that a common point of $AA'$ and $BB'$ lies on the altitude of the triangle dropped from the vertex $C.$

2010 Sharygin Geometry Olympiad First Round p21 grades 10-11

A given convex quadrilateral $ABCD$ is such that $\angle ABD + \angle ACD > \angle BAC + \angle BDC.$ Prove that $S_{ABD}+S_{ACD} > S_{BAC}+S_{BDC}.$

2010 Sharygin Geometry Olympiad First Round p22 grades 10-11

A circle centered at a point $F$ and a parabola with focus $F$ have two common points. Prove that there exist four points $A, B, C, D$ on the circle such that the lines $AB, BC, CD$ and $DA$ touch the parabola.

2010 Sharygin Geometry Olympiad First Round p23 grades 10-11

A cyclic hexagon $ABCDEF$ is such that $AB \cdot CF= 2BC \cdot FA, CD \cdot EB = 2 DE \cdot BC$ and $EF \cdot AD = 2FA \cdot DE.$ Prove that the lines $AD, BE$ and $CF$ are concurrent.

2010 Sharygin Geometry Olympiad First Round p24 grades 10-11

Let us have a line $\ell$ in the space and a point $A$ not lying on $\ell.$ For an arbitrary line $\ell'$ passing through $A$, $XY$ ($Y$ is on $\ell'$) is a common perpendicular to the lines $\ell$ and $\ell'.$ Find the locus of points $Y.$

2010 Sharygin Geometry Olympiad First Round p25 grade 11

For two different regular icosahedrons it is known that some six of their vertices are vertices of a regular octahedron. Find the ratio of the edges of these icosahedrons.

2009-2010  Final Round

grade 8

2010 Sharygin Geometry Olympiad Finals 8.1

In the non-isosceles triangle $ABC$ constructed are the  altitude from the vertex $A$ and angle bisectors from the other two vertices. Prove that the circumcircle of the triangle formed by these three lines, is tangent to the angle bisector  from the vertex $A$.

2010 Sharygin Geometry Olympiad Finals 8.2

Given two points $A$ and $B$. Find the locus of points $C$ such that points $A, B$ and $C$  can be covered by circle with radius $1$.

by Arseny Akopyan

2010 Sharygin Geometry Olympiad Finals 8.3

In a convex quadrilateral $ABCD$, rays $AB$ and $DC$ intersect at point $K$. On the bisector of the angle $AKD$, let  $P$ be a point such that the lines $BP$ and $CP$ bisect the segments $AC$ and $BD$ respectively. Prove that $AB = CD$.

2010 Sharygin Geometry Olympiad Finals 8.4

In equal angles $X_1OY$ and $YOX_2$ the circles $\omega_1$ and $\omega_2$ are inscribed, touching the sides $OX_1$ and $OX_2$ at points $A_1$ and $A_2$, respectively, and the sides $OY$ at points $B_1$ and $B_2$. Point $C_1$, is  the second point the intersection of $A_1B_2$ and $\omega_1$, and the point $C_2$ is the second intersection point of $A_2B_1$ and $\omega_2$. Prove that $C_1C_2$ is a common tangent to the circles.

2010 Sharygin Geometry Olympiad Finals 8.5

The altitude $AH$, the bisector $BL$, and the median $CM$ are drawn in the triangle $ABC$. It is known that in the triangle $HLM$, the straight line $AH$ is the altitude , and $BL$ is the bisector. Prove that $CM$ is in this triangle is the median.

2010 Sharygin Geometry Olympiad Finals 8.6

Points $E, F$ are midpoints of sides $BC, CD$ of square $ABCD$. Lines $AE$ and $BF$ intersect at point $P$. Prove that $\angle PDA = \angle AED$.

2010 Sharygin Geometry Olympiad Finals 8.7

Each of the two regular polygons $P$ and $Q$ was cut by a straight line into two parts. One of the parts $P$ and one of the parts $Q$ are folded together along the cut line. Can it happen a regular polygon not equal to one of the original, and if so, how many sides can it have?

2010 Sharygin Geometry Olympiad Finals 8.8 

Bisectors $AA_1$ and $BB_1$ of triangle $ABC$ intersect at point $I$. With bases on segments $A_1I$ and $B_1I$ are constructed isosceles triangles with tops $A_2$ and $B_2$ lying on the line $AB$. It is known that the straight line $CI$ divides the segment $A_2B_2$ in half. Is it true that the triangle $ABC$ is isosceles?

grade 9

2010 Sharygin Geometry Olympiad Finals 9.1

For each vertex of the triangle $ABC$, we found the angle between the altitude and angle bisector drawn from this vertex. It turned out that these angles at the vertices $A$ and $B$ are equal to each other and less than the angle at the vertex $C$. What is the angle $C$ of the triangle?

2010 Sharygin Geometry Olympiad Finals 9.2

Two triangles intersect. Prove that inside the circumcircle of one triangle lies at least one vertex of the other. (Here, the triangle is considered the part of the plane bounded by a closed three-part broken line,  a point lying on a circle is considered to be lying inside it.)

2010 Sharygin Geometry Olympiad Finals 9.3

On a line lie points $X, Y, Z$ (in that order).Equilateral triangles $XAB,YAB,YCD$ , have the vertices of the first and third oriented counterclockwise, and second clockwise. Prove that $AC,BD$ and $XY$ intersect at one point.

by V.Α. Yasinsky

2010 Sharygin Geometry Olympiad Finals 9.4

 In the triangle $ABC$ we marked points $A', B'$  touchpoints of the sides $BC, AC$ with the inscribed circle and the intersection point $G$ of the segments $AA'$ and $BB'$. After that, the triangle itself was erased. Restore it with a compass and a ruler.

2010 Sharygin Geometry Olympiad Finals 9.5

Circle inscribed in right triangle $ABC$ ($\angle ABC =90^o$), touches the sides $AB, BC, AC$ at points $C_1, A_1, B_1$, respectively. $A$- excircle touches the side $BC$ at point $A_2$. $A_0$ is the center of the circumcircle of triangle $A_1A_2B_1$, the point $C_0$ is defined similarly. Find the angle $A_0BC_0$.

2010 Sharygin Geometry Olympiad Finals 9.6
An arbitrary straight line passing through the vertex $B$ of a triangle $ABC$ intersects side $AC$ at point $K$, and the circumscribed circle at point $M$. Find the locus of the centers of the cirumcircles of triangles $AMK$.

2010 Sharygin Geometry Olympiad Finals 9.7

In the triangle $ABC$, $AL_a$ and $AM_a$ are the inner and outer bisectors of angle $A$ respectively. Let $\omega_a$ be a circle symmetrical to the circumcircle of triangle $AL_aM_a$ relative to the center $BC$. The circle $\omega_b$ is defined similarly. Prove that $\omega_a$ and $\omega_b$ are tangent if and only if when triangle $ABC$ is  right.

2010 Sharygin Geometry Olympiad Finals 9.8

A regular polygon is drawn on the blackboard. Volodya wants to mark $k$ points on his perimeter so that there is no other regular polygon (not necessarily with the same number of sides), also containing marked points on its perimeter. Find the smallest $k$, enough for any initial polygon.

grade 10 

2010 Sharygin Geometry Olympiad Finals 10.1

In a right triangle, let $O, I$ be the centers of the circumscribed and inscribed circles of triangles, $R, r$ are the radii of these circles, $J$ is a point symmetric of the vertex of a right angle wrt $I$. Find $OJ$.

2010 Sharygin Geometry Olympiad Finals 10.2

Each of two equal circles $\omega_1$ and $\omega_2$ passes through the center of the other. The triangle $ABC$ is inscribed in $\omega_1$, and the lines $AC, BC$ are tangent to $\omega_2$. Prove that $cos \angle A + cos \angle B = 1$.

2010 Sharygin Geometry Olympiad Finals 10.3

Two convex polygons $A_1A_2...A_n$ and $B_1B_2...B_n$ ($n\ge 4$) are such that any side of the first is larger than the corresponding side of the second. Could it be that any diagonal of the second is more than the corresponding diagonal of the first?

2010 Sharygin Geometry Olympiad Finals 10.4

The projections of two points on the sides of a quadrilateral lie on two different concentric circles (the projections of each point form an inscribed quadrilateral, and the radii of the respective circles are different). Prove that the quadrilateral is a parallelogram.

2010 Sharygin Geometry Olympiad Finals 10.5

In the right triangle $ABC$ ($\angle B = 90^o$)  let $BH$ be the altitude.The circle inscribed in the triangle $ABH$ touches the sides $AB,  AH$ in points $H_1, B_1$, respectively, the circle inscribed in the triangle $CBH$ touches the sides $CB, CH$ in points $H_2, B_2$, respectively. Let $O$ be the center of the circumscribed circle of the triangle $H_1BH_2$. Prove that $OB_1 = OB_2$.

2010 Sharygin Geometry Olympiad Finals 10.6

The inscribed circle of a triangle $ABC$ touches its sides at points $A', B'$ and $C'$. It is known that the orthocenters of triangles $ABC$ and $A'B'C'$ coincide. Is it true that $ABC$ is right?

2010 Sharygin Geometry Olympiad Finals 10.7

Each of the two regular polyhedra $P$ and $Q$ was cut by a plane into two parts. 
One of the parts $P$ and one of the parts $Q$ was applied to each other along the section plane. 
Can we get a regular polyhedron not equal to any of the original, and if so, how many edges can it have?

2010 Sharygin Geometry Olympiad Finals 10.8

 Around the triangle $ABC$ is circumscribed circle $k$. On the sides of the triangle We marked three points $A_1, B_1$ and $C_1$, after which the triangle itself was erased. Prove that it can be uniquely recovered if and only if straight lines $AA_1, BB_1$ and $CC_1$ intersect in one the point.

2010-2011 First Round 

2011 Sharygin Geometry Olympiad First Round p1 grade 8 

Does a convex heptagon exist which can be divided into 2011 equal triangles?

2011 Sharygin Geometry Olympiad First Round p2 grade 8 

Let $ABC$ be a triangle with sides $AB = 4$ and $AC = 6$. Point $H$ is the projection of vertex $B$ to the bisector of angle $A$. Find $MH$, where $M$ is the midpoint of $BC$.

2011 Sharygin Geometry Olympiad First Round p3 grade 8 

Let $ABC$ be a triangle with $\angle{A} = 60^\circ$. The midperpendicular of segment $AB$ meets line $AC$ at point $C_1$. The midperpendicular of segment $AC$ meets line $AB$ at point $B_1$. Prove that line $B_1C_1$ touches the incircle of triangle $ABC$.

2011 Sharygin Geometry Olympiad First Round p4 grade 8 

Segments $AA'$, $BB'$, and $CC'$ are the bisectrices of triangle $ABC$. It is known that these lines are also the bisectrices of triangle $A'B'C'$. Is it true that triangle $ABC$ is regular?

2011 Sharygin Geometry Olympiad First Round p5 grade 8 

Given triangle $ABC$. The midperpendicular of side $AB$ meets one of the remaining sides at point $C'$. Points $A'$ and $B'$ are defined similarly. Find all triangles $ABC$ such that triangle $A'B'C'$ is regular.

2011 Sharygin Geometry Olympiad First Round p6 grade 8 

Two unit circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. $M$ is an arbitrary point of $\omega_1$, $N$ is an arbitrary point of $\omega_2$. Two unit circles $\omega_3$ and $\omega_4$ pass through both points $M$ and $N$. Let $C$ be the second common point of $\omega_1$ and $\omega_3$, and $D$ be the second common point of $\omega_2$ and $\omega_4$. Prove that $ACBD$ is a parallelogram.

2011 Sharygin Geometry Olympiad First Round p7 grades 8-9

Points $P$ and $Q$ on sides $AB$ and $AC$ of triangle $ABC$ are such that $PB = QC$. Prove that $PQ < BC$.

2011 Sharygin Geometry Olympiad First Round p8 grades 8-9
The incircle of right-angled triangle $ABC$ ($\angle B = 90^o$) touches $AB,BC,CA$ at points $C_1,A_1,B_1$ respectively. Points $A_2, C_2$ are the reflections of $B_1$ in lines $BC, AB$ respectively. Prove that lines $A_1A_2$ and $C_1C_2$ meet on the median of triangle $ABC$.

2011 Sharygin Geometry Olympiad First Round p9 grades 8-9

Let $H$ be the orthocenter of triangle $ABC$. The tangents to the circumcircles of triangles $CHB$ and $AHB$ at point $H$ meet $AC$ at points $A_1$ and $C_1$ respectively. Prove that $A_1H = C_1H$.

2011 Sharygin Geometry Olympiad First Round p10 grades 8-9

The diagonals of trapezoid $ABCD$ meet at point $O$. Point $M$ of lateral side $CD$ and points $P, Q$ of bases $BC$ and $AD$ are such that segments $MP$ and $MQ$ are parallel to the diagonals of the trapezoid. Prove that line $PQ$ passes through point $O$.

2011 Sharygin Geometry Olympiad First Round p11 grades 8-10 

The excircle of right-angled triangle $ABC$ ($\angle B =90^o$) touches side $BC$ at point $A_1$ and touches line $AC$ in point $A_2$. Line $A_1A_2$ meets the incircle of $ABC$ for the first time at point $A'$, point $C'$ is defined similarly. Prove that $AC||A'C'$.

2011 Sharygin Geometry Olympiad First Round p12 grades 8-10 

Let $AP$ and $BQ$ be the altitudes of acute-angled triangle $ABC$. Using a compass and a ruler, construct a point $M$ on side $AB$ such that $\angle AQM = \angle BPM$.

2011 Sharygin Geometry Olympiad First Round p13 grades 8-11

a) Find the locus of centroids for triangles whose vertices lie on the sides of a given triangle (each side contains a single vertex).
b)  Find the locus of centroids for tetrahedrons whose vertices lie on the faces of a given tetrahedron (each face contains a single vertex).

2011 Sharygin Geometry Olympiad First Round p14 grade 9  

In triangle $ABC$, the altitude and the median from vertex $A$ form (together with line $BC$) a triangle such that the bisectrix of angle $A$ is the median; the altitude and the median from vertex $B$ form (together with line AC) a triangle such that the bisectrix of angle $B$ is the bisectrix. Find the ratio of sides for triangle $ABC$.

2011 Sharygin Geometry Olympiad First Round p15 grades 9-10 
Given a circle with center $O$ and radius equal to $1$. $AB$ and $AC$ are the tangents to this circle from point $A$. Point $M$ on the circle is such that the areas of quadrilaterals $OBMC$ and $ABMC$ are equal. Find $MA$.

2011 Sharygin Geometry Olympiad First Round p16 grades 9-10   

Given are triangle $ABC$ and line $\ell$. The reflections of $\ell$ in $AB$ and $AC$ meet at point $A_1$. Points $B_1, C_1$ are defined similarly. Prove that
a) lines $AA_1, BB_1, CC_1$ concur,
b) their common point lies on the circumcircle of $ABC$
c) two points constructed in this way for two perpendicular lines are opposite.

2011 Sharygin Geometry Olympiad First Round p17 grades 9-11 

a) Does there exist a triangle in which the shortest median is longer that the longest bisectrix?
b) Does there exist a triangle in which the shortest bisectrix is longer that the longest altitude?

  

2011 Sharygin Geometry Olympiad First Round p18 grades 9-11  

On the plane, given are $n$ lines in general position, i.e. any two of them aren’t parallel and any three of them don’t concur. These lines divide the plane into several parts. What is
a) the minimal,
b) the maximal number of these parts that can be angles?

2011 Sharygin Geometry Olympiad First Round p19 grades 9-11

Does there exist a nonisosceles triangle such that the altitude from one vertex, the bisectrix from the second one and the median from the third one are equal?

2011 Sharygin Geometry Olympiad First Round p20 grades 9-11 

Quadrilateral $ABCD$ is circumscribed around a circle with center $I$. Points $M$ and $N$ are the midpoints of diagonals $AC$ and $BD$. Prove that $ABCD$ is cyclic quadrilateral if and only if $IM : AC = IN : BD$.

2011 Sharygin Geometry Olympiad First Round p21 grades 10-11  

On a circle with diameter $AC$, let $B$ be an arbitrary point distinct from $A$ and $C$. Points $M, N$ are the midpoints of chords $AB, BC$, and points $P, Q$ are the midpoints of smaller arcs restricted by these chords. Lines $AQ$ and $BC$ meet at point $K$, and lines $CP$ and $AB$ meet at point $L$. Prove that lines $MQ, NP$ and $KL$ concur.

2011 Sharygin Geometry Olympiad First Round p22 grades 10-11  

Let $CX, CY$ be the tangents from vertex $C$ of triangle $ABC$ to the circle passing through the midpoints of its sides. Prove that lines $XY , AB$ and the tangent to the circumcircle of $ABC$ at point $C$ concur.

2011 Sharygin Geometry Olympiad First Round p23 grades 10-11 

Given are triangle $ABC$ and line $\ell$ intersecting $BC, CA$ and $AB$ at points $A_1, B_1$ and $C_1$ respectively. Point $A'$ is the midpoint of the segment between the projections of $A_1$ to $AB$ and $AC$. Points $B'$ and $C'$ are defined similarly.
(a) Prove that $A', B'$ and $C'$ lie on some line $\ell'$.
(b) Suppose $\ell'$  passes through the circumcenter of $\triangle ABC$. Prove that in this case $\ell'$ passes through the center of its nine-points circle.

2011 Sharygin Geometry Olympiad First Round p24 grades 10-11 

Given is an acute-angled triangle $ABC$. On sides $BC, CA, AB$, find points $A', B', C'$ such that the longest side of triangle $A'B'C'$ is minimal.

2011 Sharygin Geometry Olympiad First Round p25 grades 10-11 

Three equal regular tetrahedrons have the common center. Is it possible that all faces of the polyhedron that forms their intersection are equal?

2010-2011 Final Round

grade 8

2011 Sharygin Geometry Olympiad Finals 8.1

The diagonals of a trapezoid are perpendicular, and its altitude is equal to the medial line. Prove that this trapezoid is isosceles.

2011 Sharygin Geometry Olympiad Finals 8.2

Peter made a paper rectangle, put it on an identical rectangle and pasted both rectangles along their perimeters. Then he cut the upper rectangle along one of its diagonals and along the perpendiculars to this diagonal from two remaining vertices. After this he turned back the obtained triangles in such a way that they, along with the lower rectangle form a new rectangle.
Let this new rectangle be given. Restore the original rectangle using compass and ruler.

2011 Sharygin Geometry Olympiad Finals 8.3

The line passing through vertex $A$ of triangle $ABC$ and parallel to $BC$ meets the circumcircle of $ABC$ for the second time at point $A_1$. Points $B_1$ and $C_1$ are defined similarly. Prove that the perpendiculars from $A_1, B_1, C_1$ to $BC, CA, AB$ respectively concur.

2011 Sharygin Geometry Olympiad Finals 8.4

Given the circle of radius $1$ and several its chords with the sum of lengths $1$. Prove that one can be inscribe a regular hexagon into that circle so that its sides don’t intersect those chords.

2011 Sharygin Geometry Olympiad Finals 8.5

A line passing through vertex $A$ of regular triangle $ABC$ doesn’t intersect segment $BC$. Points $M$ and $N$ lie on this line, and $AM = AN = AB$ (point $B$ lies inside angle $MAC$). Prove that the quadrilateral formed by lines $AB, AC, BN, CM$ is cyclic.

2011 Sharygin Geometry Olympiad Finals 8.6 

Let $BB_1$ and $CC_1$ be the altitudes of acute-angled triangle $ABC$, and $A_0$ is the midpoint of $BC$. Lines $A_0B_1$ and $A_0C_1$ meet the line passing through $A$ and parallel to $BC$ in points $P$ and $Q$. Prove that the incenter of triangle $PA_0Q$ lies on the altitude of triangle $ABC$.

2011 Sharygin Geometry Olympiad Finals 8.7 

Let a point $M$ not lying on coordinates axes be given. Points $Q$ and $P$ move along $Y$ - and $X$-axis respectively so that angle $P M Q$ is always right. Find the locus of points symmetric to $M$ wrt $P Q$.

2011 Sharygin Geometry Olympiad Finals 8.8

Using only the ruler, divide the side of a square table into $n$ equal parts.
All lines drawn must lie on the surface of the table.

grade 9

2011 Sharygin Geometry Olympiad Finals 9.1

Altitudes $AA_1$ and $BB_1$ of triangle ABC meet in point $H$. Line $CH$ meets the semicircle with diameter $AB$, passing through $A_1, B_1$, in point $D$. Segments $AD$ and $BB_1$ meet in point $M$, segments $BD$ and $AA_1$ meet in point $N$. Prove that the circumcircles of triangles $B_1DM$ and $A_1DN$ touch.

2011 Sharygin Geometry Olympiad Finals 9.2

In triangle $ABC, \angle B = 2\angle C$. Points $P$ and $Q$ on the medial perpendicular to $CB$ are such that $\angle CAP = \angle PAQ = \angle QAB = \frac{\angle A}{3}$ . Prove that $Q$ is the circumcenter of triangle $CPB$.

2011 Sharygin Geometry Olympiad Finals 9.3

Restore the isosceles triangle $ABC$ ($AB = AC$) if the common points $I, M, H$ of bisectors, medians and altitudes respectively are given.

2011 Sharygin Geometry Olympiad Finals 9.4

Quadrilateral $ABCD$ is inscribed into a circle with center $O$. The bisectors of its angles form a cyclic quadrilateral with circumcenter $I$, and its external bisectors form a cyclic quadrilateral with circumcenter $J$. Prove that $O$ is the midpoint of $IJ$.

2011 Sharygin Geometry Olympiad Finals 9.5

It is possible to compose a triangle from the altitudes of a given triangle. Can we conclude that it is possible to compose a triangle from its bisectors?

2011 Sharygin Geometry Olympiad Finals 9.6

In triangle $ABC$ $AA_0$ and $BB_0$ are medians, $AA_1$ and $BB_1$ are altitudes. The circumcircles of triangles $CA_0B_0$ and $CA_1B_1$ meet again in point $M_c$. Points $M_a, M_b$ are defined similarly. Prove that points $M_a, M_b, M_c$ are collinear and lines $AM_a, BM_b, CM_c$ are parallel.

2011 Sharygin Geometry Olympiad Finals 9.7

Circles $\omega$  and $\Omega$ are inscribed into the same angle. Line $\ell$ meets the sides of angles, $\omega$  and $\Omega$  in points $A$ and $F, B$ and $C, D$ and $E$ respectively (the order of points on the line is $A,B,C,D,E, F$). It is known that$BC = DE$. Prove that $AB = EF$.

2011 Sharygin Geometry Olympiad Finals 9.8

A convex $n$-gon $P$, where $n > 3$, is dissected into equal triangles by diagonals non-intersecting inside it. Which values of $n$ are possible, if $P$ is circumscribed?

grade 10 

2011 Sharygin Geometry Olympiad Finals 10.1

In triangle $ABC$ the midpoints of sides $AC, BC$, vertex $C$ and the centroid lie on the same circle. Prove that this circle touches the circle passing through $A, B$ and the orthocenter of triangle $ABC$.

2011 Sharygin Geometry Olympiad Finals 10.2

Quadrilateral $ABCD$ is circumscribed. Its incircle touches sides $AB, BC, CD, DA$ in points $K, L, M, N$ respectively. Points $A', B', C', D'$ are the midpoints of segments $LM, MN, NK, KL$. Prove that the quadrilateral formed by lines $AA', BB', CC', DD'$ is cyclic.

2011 Sharygin Geometry Olympiad Finals 10.3

Given two tetrahedrons $A_1A_2A_3A_4$ and $B_1B_2B_3B_4$. Consider six pairs of edges $A_iA_j$ and $B_kB_l$, where ($i, j, k, l$) is a transposition of numbers ($1, 2, 3, 4$) (for example $A_1A_2$ and $B_3B_4$). It is known that for all but one such pairs the edges are perpendicular. Prove that the edges in the remaining pair also are perpendicular.

2011 Sharygin Geometry Olympiad Finals 10.4

Point $D$ lies on the side $AB$ of triangle $ABC$. The circle inscribed in angle $ADC$ touches internally the circumcircle of triangle $ACD$. Another circle inscribed in angle $BDC$ touches internally the circumcircle of triangle $BCD$. These two circles touch segment $CD$ in the same point $X$. Prove that the perpendicular from $X$ to $AB$ passes through the incenter of triangle $ABC$

2011 Sharygin Geometry Olympiad Finals 10.5

The touching point of the excircle with the side of a triangle and the base of the altitude to this side are symmetric wrt the base of the corresponding bisector. Prove that this side is equal to one third of the perimeter.

2011 Sharygin Geometry Olympiad Finals 10.6 

Prove that for any nonisosceles triangle $l_1^2>\sqrt3 S>l_2^2$, where $l_1, l_2$ are the greatest and the smallest bisectors of the triangle and $S$ is its area.

2011 Sharygin Geometry Olympiad Finals 10.7

Point $O$ is the circumcenter of acute-angled triangle $ABC$, points $A_1,B_1, C_1$ are the bases of its altitudes. Points $A', B', C'$ lying on lines $OA_1, OB_1, OC_1$ respectively are such that quadrilaterals $AOBC', BOCA', COAB'$ are cyclic. Prove that the circumcircles of triangles $AA_1A', BB_1B', CC_1C'$ have a common point.

2011 Sharygin Geometry Olympiad Finals 10.8

Given a sheet of tin $6\times 6$. It is allowed to bend it and to cut it but in such a way that it doesn’t fall to pieces. How to make a cube with edge $2$, divided by partitions into unit cubes?

2011-2012 First Round

2012 Sharygin Geometry Olympiad First Round p1 grade 8 

In triangle $ABC$ point $M$ is the midpoint of side $AB$, and point $D$ is the foot of altitude $CD$. Prove that $\angle A = 2\angle B$ if and only if $AC = 2 MD$

2012 Sharygin Geometry Olympiad First Round p2 grade 8 

A cyclic $n$-gon is divided by non-intersecting (inside the $n$-gon) diagonals to $n-2$ triangles. Each of these triangles is similar to at least one of the remaining ones. For what $n$ this is possible?

2012 Sharygin Geometry Olympiad First Round p3 grade 8 

A circle with center $I$ touches sides $AB,BC,CA$ of triangle $ABC$ in points $C_{1},A_{1},B_{1}$. Lines $AI, CI, B_{1}I$ meet $A_{1}C_{1}$ in points $X, Y, Z$ respectively. Prove that $\angle Y B_{1}Z = \angle XB_{1}Z$.

2012 Sharygin Geometry Olympiad First Round p4 grade 8

Given triangle $ABC$. Point $M$ is the midpoint of side $BC$, and point $P$ is the projection of $B$ to the perpendicular bisector of segment $AC$. Line $PM$ meets $AB$ in point $Q$. Prove that triangle $QPB$ is isosceles.

2012 Sharygin Geometry Olympiad First Round p5 grade 8 

On side $AC$ of triangle $ABC$ an arbitrary point is selected $D$. The tangent in $D$ to the circumcircle of triangle $BDC$ meets $AB$ in point $C_{1}$; point $A_{1}$ is defined similarly. Prove that $A_{1}C_{1}\parallel AC$.

2012 Sharygin Geometry Olympiad First Round p6 grades 8-9 

Point $C_{1}$ of hypothenuse $AC$ of a right-angled triangle $ABC$ is such that $BC = CC_{1}$. Point $C_{2}$ on cathetus $AB$ is such that $AC_{2} = AC_{1}$; point $A_{2}$ is defined similarly. Find angle $AMC$, where $M$ is the midpoint of $A_{2}C_{2}$.

2012 Sharygin Geometry Olympiad First Round p7 grades 8-9 
In a non-isosceles triangle $ABC$ the bisectors of angles $A$ and $B$ are inversely proportional to the respective sidelengths. Find angle $C$.

2012 Sharygin Geometry Olympiad First Round p8 grades 8-9 

Let $BM$ be the median of right-angled triangle $ABC (\angle B = 90^{\circ})$. The incircle of triangle $ABM$ touches sides $AB, AM$ in points $A_{1},A_{2}$; points $C_{1}, C_{2}$ are defined similarly. Prove that lines $A_{1}A_{2}$ and $C_{1}C_{2}$ meet on the bisector of angle $ABC$.

2012 Sharygin Geometry Olympiad First Round p9 grades 8-9 

In triangle $ABC$, given lines $l_{b}$ and $l_{c}$ containing the bisectors of angles $B$ and $C$, and the foot $L_{1}$ of the bisector of angle $A$. Restore triangle $ABC$.

2012 Sharygin Geometry Olympiad First Round p10 grades 8-11 

In a convex quadrilateral all sidelengths and all angles are pairwise different.
a) Can the greatest angle be adjacent to the greatest side and at the same time the smallest angle be adjacent to the smallest side?
b) Can the greatest angle be non-adjacent to the smallest side and at the same time the smallest angle be non-adjacent to the greatest side?

2012 Sharygin Geometry Olympiad First Round p11 grades 8-11 

Given triangle $ABC$ and point $P$. Points $A', B', C'$ are the projections of $P$ to $BC, CA, AB$. A line passing through $P$ and parallel to $AB$ meets the circumcircle of triangle $PA'B'$ for the second time in point $C_{1}$. Points $A_{1}, B_{1}$ are defined similarly. Prove that
a) lines $AA_{1}, BB_{1}, CC_{1}$ concur;
b) triangles $ABC$ and $A_{1}B_{1}C_{1}$ are similar.

2012 Sharygin Geometry Olympiad First Round p12 grades 9-10 

Let $O$ be the circumcenter of an acute-angled triangle $ABC$. A line passing through $O$ and parallel to $BC$ meets $AB$ and $AC$ in points $P$ and $Q$ respectively. The sum of distances from $O$ to $AB$ and $AC$ is equal to $OA$. Prove that $PB + QC = PQ$.

2012 Sharygin Geometry Olympiad First Round p13 grades 9-10 

Points $A, B$ are given. Find the locus of points $C$ such that $C$, the midpoints of $AC, BC$ and the centroid of triangle $ABC$ are concyclic.

2012 Sharygin Geometry Olympiad First Round p14 grades 9-10 

In a convex quadrilateral $ABCD$ suppose $AC \cap BD = O$ and $M$ is the midpoint of $BC$. Let $MO \cap AD = E$. Prove that $\frac{AE}{ED} = \frac{S_{\triangle ABO}}{S_{\triangle CDO}}$.

2012 Sharygin Geometry Olympiad First Round p15 grades 9-11 

Given triangle $ABC$. Consider lines $l$ with the next property: the reflections of $l$ in the sidelines of the triangle concur. Prove that all these lines have a common point.

2012 Sharygin Geometry Olympiad First Round p16 grades 9-11 

Given right-angled triangle $ABC$ with hypothenuse $AB$. Let $M$ be the midpoint of $AB$ and $O$ be the center of circumcircle $\omega$ of triangle $CMB$. Line $AC$ meets $\omega$ for the second time in point $K$. Segment $KO$ meets the circumcircle of triangle $ABC$ in point $L$. Prove that segments $AL$ and $KM$ meet on the circumcircle of triangle $ACM$.

2012 Sharygin Geometry Olympiad First Round p17 grades 9-11 

A square $ABCD$ is inscribed into a circle. Point $M$ lies on arc $BC$, $AM$ meets $BD$ in point $P$, $DM$ meets $AC$ in point $Q$. Prove that the area of quadrilateral $APQD$ is equal to the half of the area of the square.

2012 Sharygin Geometry Olympiad First Round p18 grades 9-11 

A triangle and two points inside it are marked. It is known that one of the triangle’s angles is equal to $58^{\circ}$, one of two remaining angles is equal to $59^{\circ}$, one of two given points is the incenter of the triangle and the second one is its circumcenter. Using only the ruler without partitions determine where is each of the angles and where is each of the centers.

2012 Sharygin Geometry Olympiad First Round p19 grades 10-11 

Two circles with radii 1 meet in points $X, Y$, and the distance between these points also is equal to $1$. Point $C$ lies on the first circle, and lines $CA, CB$ are tangents to the second one. These tangents meet the first circle for the second time in points $B', A'$. Lines $AA'$ and $BB'$ meet in point $Z$. Find angle $XZY$.

2012 Sharygin Geometry Olympiad First Round p20 grades 10-11 

Point $D$ lies on side $AB$ of triangle $ABC$. Let $\omega_1$ and $\Omega_1,\omega_2$ and $\Omega_2$ be the incircles and the excircles (touching segment $AB$) of triangles $ACD$ and $BCD.$ Prove that the common external tangents to $\omega_1$ and $\omega_2,$ $\Omega_1$ and $\Omega_2$ meet on $AB$.

2012 Sharygin Geometry Olympiad First Round p21 grades 10-11 

Two perpendicular lines pass through the orthocenter of an acute-angled triangle. The sidelines of the triangle cut on each of these lines two segments: one lying inside the triangle and another one lying outside it. Prove that the product of two internal segments is equal to the product of two external segments.

2012 Sharygin Geometry Olympiad First Round p22 grades 10-11 

A circle $\omega$ with center $I$ is inscribed into a segment of the disk, formed by an arc and a chord $AB$. Point $M$ is the midpoint of this arc $AB$, and point $N$ is the midpoint of the complementary arc. The tangents from $N$ touch $\omega$ in points $C$ and $D$. The opposite sidelines $AC$ and $BD$ of quadrilateral $ABCD$ meet in point $X$, and the diagonals of $ABCD$ meet in point $Y$. Prove that points $X, Y, I$ and $M$ are collinear.

2012 Sharygin Geometry Olympiad First Round p23 grades 10-11 

An arbitrary point is selected on each of twelve diagonals of the faces of a cube.The centroid of these twelve points is determined. Find the locus of all these centroids.

2012 Sharygin Geometry Olympiad First Round p24 grades 10-11 

Given are $n$ $(n > 2)$ points on the plane such that no three of them are collinear. In how many ways this set of points can be divided into two non-empty subsets with non-intersecting convex envelops?

2011-2012 Final Round

grade 8

2012 Sharygin Geometry Olympiad Finals 8.1

Let $M$ be the midpoint of the base $AC$ of an acute-angled isosceles triangle $ABC$. Let $N$ be the reflection of $M$ in $BC$. The line parallel to $AC$ and passing through $N$ meets $AB$ at point $K$. Determine the value of $\angle AKC$.

by A.Blinkov 

2012 Sharygin Geometry Olympiad Finals 8.2

In a triangle $ABC$ the bisectors $BB'$ and $CC'$ are drawn. After that, the whole picture except the points $A, B'$, and $C'$ is erased. Restore the triangle using a compass and a ruler.

by A.Karlyuchenko 

2012 Sharygin Geometry Olympiad Finals 8.3

A paper square was bent by a line in such way that one vertex came to a side not containing this vertex. Three circles are inscribed into three obtained triangles (see Figure). Prove that one of their radii is equal to the sum of the two remaining ones.

by L.Steingarts 

2012 Sharygin Geometry Olympiad Finals 8.4

Let $ABC$ be an isosceles triangle with $\angle B = 120^o$ . Points $P$ and $Q$ are chosen on the prolongations of segments $AB$ and $CB$ beyond point $B$ so that the rays $AQ$ and $CP$ intersect and are perpendicular to each other. Prove that $\angle PQB = 2\angle PCQ$.

by A.Akopyan, D.Shvetsov 

2012 Sharygin Geometry Olympiad Finals 8.5

Do there exist a convex quadrilateral and a point $P$ inside it such that the sum of distances from $P$ to the vertices of the quadrilateral is greater than its perimeter?

by A.Akopyan 

2012 Sharygin Geometry Olympiad Finals 8.6

Let $\omega$ be the circumcircle of triangle $ABC$. A point $B_1$ is chosen on the prolongation of side $AB$ beyond point B so that $AB_1 = AC$. The angle bisector of $\angle BAC$ meets $\omega$  again at point $W$. Prove that the orthocenter of triangle $AWB_1$ lies on $\omega$ .

by A.Tumanyan 

2012 Sharygin Geometry Olympiad Finals 8.7 

The altitudes $AA_1$ and $CC_1$ of an acute-angled triangle $ABC$ meet at point $H$. Point $Q$ is the reflection of the midpoint of $AC$ in line $AA_1$, point $P$ is the midpoint of segment $A_1C_1$. Prove that $\angle QPH = 90^o$.

by D.Shvetsov 

2012 Sharygin Geometry Olympiad Finals 8.8

 A square is divided into several (greater than one) convex polygons with mutually different numbers of sides. Prove that one of these polygons is a triangle.

by A.Zaslavsky

grade 9

2012 Sharygin Geometry Olympiad Finals 9.1

The altitudes $AA_1$ and $BB_1$ of an acute-angled triangle ABC meet at point $O$. Let $A_1A_2$ and $B_1B_2$ be the altitudes of triangles $OBA_1$ and $OAB_1$ respectively. Prove that $A_2B_2$ is parallel to $AB$.

by L.Steingarts 

2012 Sharygin Geometry Olympiad Finals 9.2 

Three parallel lines passing through the vertices $A, B$, and $C$ of triangle $ABC$ meet its circumcircle again at points $A_1, B_1$, and $C_1$ respectively. Points $A_2, B_2$, and $C_2$ are the reflections of points $A_1, B_1$, and $C_1$  in $BC, CA$, and $AB$ respectively. Prove that the lines $AA_2, BB_2, CC_2$ are concurrent.

by D.Shvetsov, A.Zaslavsky  

2012 Sharygin Geometry Olympiad Finals 9.3

In triangle $ABC$, the bisector $CL$ was drawn. The incircles of triangles $CAL$ and $CBL$ touch $AB$ at points $M$ and $N$ respectively. Points $M$ and $N$ are marked on the picture, and then the whole picture except the points $A, L, M$, and $N$ is erased. Restore the triangle using a compass and a ruler.

by V.Protasov 

2012 Sharygin Geometry Olympiad Finals 9.4

 Determine all integer $n > 3$ for which a regular $n$-gon can be divided into equal triangles by several (possibly intersecting) diagonals.

by B.Frenkin

2012 Sharygin Geometry Olympiad Finals 9.5 

Let $ABC$ be an isosceles right-angled triangle. Point $D$ is chosen on the prolongation of the hypothenuse $AB$ beyond point $A$ so that $AB = 2AD$. Points $M$ and $N$ on side $AC$ satisfy the relation $AM = NC$. Point $K$ is chosen on the prolongation of $CB$ beyond point $B$ so that $CN = BK$. Determine the angle between lines $NK$ and $DM$.

by M.Kungozhin 

2012 Sharygin Geometry Olympiad Finals 9.6

Let $ABC$ be an isosceles triangle with $BC = a$ and $AB = AC = b$. Segment $AC$ is the base of an isosceles triangle $ADC$ with $AD = DC = a$ such that points $D$ and $B$ share the opposite sides of AC. Let $CM$ and $CN$ be the bisectors in triangles $ABC$ and $ADC$ respectively. Determine the circumradius of triangle $CMN$.

by M.Rozhkova 

2012 Sharygin Geometry Olympiad Finals 9.7 

A convex pentagon $P$ is divided by all its diagonals into ten triangles and one smaller pentagon $P'$. Let $N$ be the sum of areas of five triangles adjacent to the sides of $P$ decreased by the area of $P'$. The same operations are performed with the pentagon $P'$, let $N'$ be the similar difference calculated for this pentagon. Prove that $N > N'$.

by A.Belov 

2012 Sharygin Geometry Olympiad Finals 9.8

Let $AH$ be an altitude of an acute-angled triangle $ABC$. Points $K$ and $L$ are the projections of $H$ onto sides $AB$ and $AC$. The circumcircle of $ABC$ meets line $KL$ at points $P$ and $Q$, and meets line $AH$ at points $A$ and $T$. Prove that $H$ is the incenter of triangle $PQT$.

by M.Plotnikov 

grade 10

2012 Sharygin Geometry Olympiad Finals 10.1

Determine all integer $n$ such that a surface of an $n \times n \times n$ grid cube can be pasted in one layer by paper $1 \times 2$ rectangles so that each rectangle has exactly five neighbors (by a line segment).

by A.Shapovalov 

2012 Sharygin Geometry Olympiad Finals 10.2

We say that a point inside a triangle is good if the lengths of the cevians passing through this point are inversely proportional to the respective side lengths. Find all the triangles for which the number of good points is maximal.

by A.Zaslavsky, B.Frenkin 

2012 Sharygin Geometry Olympiad Finals 10.3

 Let $M$ and $I$ be the centroid and the incenter of a scalene triangle $ABC$, and let $r$ be its inradius. Prove that $MI = r/3$ if and only if $MI$ is perpendicular to one of the sides of the triangle.

by A.Karlyuchenko

2012 Sharygin Geometry Olympiad Finals 10.4 

Consider a square. Find the locus of midpoints of the hypothenuses of rightangled triangles with the vertices lying on three different sides of the square and not coinciding with its vertices.

by B.Frenkin

2012 Sharygin Geometry Olympiad Finals 10.5

A quadrilateral $ABCD$ with perpendicular diagonals is inscribed into a circle $\omega$. Two arcs $\alpha$ and $\beta$ with diameters AB and $CD$ lie outside $\omega$. Consider two crescents formed by the circle $\omega$ and the arcs $\alpha$  and $\beta$ (see Figure). Prove that the maximal radii of the circles inscribed into these crescents are equal.

by F.Nilov 

2012 Sharygin Geometry Olympiad Finals 10.6

Consider a tetrahedron $ABCD$. A point $X$ is chosen outside the tetrahedron so that segment $XD$ intersects face $ABC$ in its interior point. Let $A' , B'$ , and $C'$ be the projections of $D$ onto the planes $XBC, XCA$, and $XAB$ respectively. Prove that $A' B'  + B' C'  + C' A'  \le  DA + DB + DC$.

by V.Yassinsky 

2012 Sharygin Geometry Olympiad Finals 10.7

Consider a triangle $ABC$. The tangent line to its circumcircle at point $C$ meets line $AB$ at point $D$. The tangent lines to the circumcircle of triangle $ACD$ at points $A$ and $C$ meet at point $K$. Prove that line $DK$ bisects segment $BC$.

by F.Ivlev

2012 Sharygin Geometry Olympiad Finals 10.8

A point $M$ lies on the side $BC$ of square $ABCD$. Let $X$, $Y$ , and $Z$ be the incenters of triangles $ABM$, $CMD$, and $AMD$ respectively. Let $H_x$, $H_y$, and $H_z$ be the orthocenters of triangles $AXB$, $CY D$, and $AZD$. Prove that $H_x$, $H_y$, and $H_z$ are collinear.

by D.Shvetsov

2012-2013 First Round

2013 Sharygin Geometry Olympiad First Round p1 grade 8

Let $ABC$ be an isosceles triangle with $AB = BC$. Point $E$ lies on the side $AB$, and $ED$ is the perpendicular from $E$ to $BC$. It is known that $AE = DE$. Find $\angle DAC$.

2013 Sharygin Geometry Olympiad First Round p2 grade 8

Let $ABC$ be an isosceles triangle ($AC = BC$) with $\angle C = 20^\circ$. The bisectors of angles $A$ and $B$ meet the opposite sides at points $A_1$ and $B_1$ respectively. Prove that the triangle $A_1OB_1$ (where $O$ is the circumcenter of $ABC$) is regular.

2013 Sharygin Geometry Olympiad First Round p3 grade 8

Let $ABC$ be a right-angled triangle ($\angle B = 90^\circ$). The excircle inscribed into the angle $A$ touches the extensions of the sides $AB$, $AC$ at points $A_1, A_2$ respectively; points $C_1, C_2$ are defined similarly. Prove that the perpendiculars from $A, B, C$ to $C_1C_2, A_1C_1, A_1A_2$ respectively, concur.

2013 Sharygin Geometry Olympiad First Round p4 grade 8

Let $ABC$ be a nonisosceles triangle. Point $O$ is its circumcenter, and point $K$ is the center of the circumcircle $w$ of triangle $BCO$. The altitude of $ABC$ from $A$ meets $w$ at a point $P$. The line $PK$ intersects the circumcircle of $ABC$ at points $E$ and $F$. Prove that one of the segments $EP$ and $FP$ is equal to the segment $PA$.

2013 Sharygin Geometry Olympiad First Round p5 grade 8

Four segments drawn from a given point inside a convex quadrilateral to its vertices, split the quadrilateral into four equal triangles. Can we assert that this quadrilateral is a rhombus?

2013 Sharygin Geometry Olympiad First Round p6 grades 8-9

Diagonals $AC$ and $BD$ of a trapezoid $ABCD$ meet at $P$. The circumcircles of triangles $ABP$ and $CDP$ intersect the line $AD$ for the second time at points  $X$ and $Y$ respectively. Let $M$ be the midpoint of segment $XY$. Prove that $BM = CM$.

2013 Sharygin Geometry Olympiad First Round p7 grades 8-9

Let $BD$ be a bisector of triangle $ABC$. Points $I_a$, $I_c$ are the incenters of triangles $ABD$, $CBD$ respectively. The line $I_aI_c$ meets $AC$ in point $Q$. Prove that $\angle DBQ = 90^\circ$.

2013 Sharygin Geometry Olympiad First Round p8 grades 8-9

Let $X$ be an arbitrary point inside the circumcircle of a triangle $ABC$. The lines $BX$ and $CX$ meet the circumcircle in points $K$ and $L$ respectively. The line $LK$ intersects $BA$ and $AC$ at points $E$ and $F$ respectively. Find the locus of points $X$ such that the circumcircles of triangles $AFK$ and $AEL$ touch.

2013 Sharygin Geometry Olympiad First Round p9 grades 8-9

Let $T_1$ and $T_2$ be the points of tangency of the excircles of a triangle $ABC$ with its sides $BC$ and $AC$ respectively. It is known that the reflection of the incenter of $ABC$ across the midpoint of $AB$ lies on the circumcircle of triangle $CT_1T_2$. Find $\angle BCA$.

2013 Sharygin Geometry Olympiad First Round p10 grades 8-9

The incircle of triangle $ABC$ touches the side $AB$ at point $C'$; the incircle of triangle $ACC'$ touches the sides $AB$ and $AC$ at points $C_1, B_1$; the incircle of triangle $BCC'$ touches the sides $AB$ and $BC$ at points $C_2$, $A_2$. Prove that the lines $B_1C_1$, $A_2C_2$, and $CC'$ concur.

2013 Sharygin Geometry Olympiad First Round p11 grades 8-9

a) Let $ABCD$ be a convex quadrilateral and $r_1 \le r_2 \le r_3 \le r_4$ be the radii of the incircles of triangles $ABC, BCD, CDA, DAB$. Can the inequality $r_4 > 2r_3$ hold?

b) The diagonals of a convex quadrilateral $ABCD$ meet in point $E$. Let $r_1 \le r_2 \le r_3 \le r_4$ be the radii of the incircles of triangles $ABE, BCE, CDE, DAE$. Can the inequality $r_2 > 2r_1$ hold?

2013 Sharygin Geometry Olympiad First Round p12 grades 8-11

On each side of triangle $ABC$, two distinct points are marked. It is known that these points are the feet of the altitudes and of the bisectors.

a) Using only a ruler determine which points are the feet of the altitudes and which points are the feet of the bisectors.

b) Solve p.a) drawing only three lines.

2013 Sharygin Geometry Olympiad First Round p13 grades 9-10

Let $A_1$ and $C_1$ be the tangency points of the incircle of triangle $ABC$ with $BC$ and $AB$ respectively, $A'$ and $C'$ be the tangency points of the excircle inscribed into the angle $B$ with the extensions of $BC$ and $AB$ respectively. Prove that the orthocenter $H$ of triangle $ABC$ lies on $A_1C_1$ if and only if the lines $A'C_1$ and $BA$ are orthogonal.

2013 Sharygin Geometry Olympiad First Round p14 grades 9-11

Let $M$, $N$ be the midpoints of diagonals $AC$, $BD$ of a right-angled trapezoid $ABCD$ ($\measuredangle A=\measuredangle D = 90^\circ$). The circumcircles of triangles $ABN$, $CDM$ meet the line $BC$ in the points $Q$, $R$. Prove that the distances from $Q$, $R$ to the midpoint of $MN$ are equal.

2013 Sharygin Geometry Olympiad First Round p15 grades 9-11

a) Triangles $A_1B_1C_1$ and $A_2B_2C_2$ are inscribed into triangle $ABC$ so that $C_1A_1 \perp BC$, $A_1B_1 \perp CA$, $B_1C_1 \perp AB$, $B_2A_2 \perp BC$, $C_2B_2 \perp CA$, $A_2C_2 \perp AB$. Prove that these triangles are equal.

b) Points $A_1$, $B_1$, $C_1$, $A_2$, $B_2$, $C_2$ lie inside a triangle $ABC$ so that $A_1$ is on segment $AB_1$, $B_1$ is on segment $BC_1$, $C_1$ is on segment $CA_1$, $A_2$ is on segment  $AC_2$, $B_2$ is on segment $BA_2$, $C_2$ is on segment $CB_2$, and the angles $BAA_1$, $CBB_2$, $ACC_1$, $CAA_2$, $ABB_2$, $BCC_2$ are equal. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are equal.

2013 Sharygin Geometry Olympiad First Round p16 grades 9-11

The incircle of triangle $ABC$ touches $BC$, $CA$, $AB$ at points $A_1$, $B_1$, $C_1$, respectively. The perpendicular from the incenter $I$ to the median from vertex $C$ meets the line $A_1B_1$ in point $K$. Prove that $CK // AB$.

2013 Sharygin Geometry Olympiad First Round p17 grades 9-11

An acute angle between the diagonals of a cyclic quadrilateral is equal to $\phi$. Prove that an acute angle between the diagonals of any other quadrilateral having the same sidelengths is smaller than $\phi$.

2013 Sharygin Geometry Olympiad First Round p18 grades 9-11

Let $AD$ be a bisector of triangle $ABC$. Points $M$ and $N$ are projections of $B$ and $C$ respectively to $AD$. The circle with diameter $MN$ intersects $BC$ at points $X$ and $Y$. Prove that $\angle BAX = \angle CAY$.

2013 Sharygin Geometry Olympiad First Round p19 grades 10-11

a) The incircle of a triangle $ABC$ touches $AC$ and $AB$ at points $B_0$ and $C_0$ respectively. The bisectors of angles $B$ and $C$ meet the perpendicular bisector to the bisector $AL$ in points $Q$ and $P$ respectively. Prove that the lines $PC_0, QB_0$ and $BC$ concur.

b) Let $AL$ be the bisector of a triangle $ABC$. Points $O_1$ and $O_2$ are the circumcenters of triangles $ABL$ and $ACL$ respectively. Points $B_1$ and $C_1$ are the projections of $C$ and $B$ to the bisectors of angles $B$ and $C$ respectively. Prove that the lines $O_1C_1, O_2B_1,$ and $BC$ concur.

c) Prove that the two points obtained in pp. a) and b) coincide.

2013 Sharygin Geometry Olympiad First Round p20 grades 10-11

Let $C_1$ be an arbitrary point on the side $AB$ of triangle $ABC$. Points $A_1$ and $B_1$ on the rays $BC$ and $AC$ are such that $\angle AC_1B_1 = \angle BC_1A_1 = \angle ACB$. The lines $AA_1$ and $BB_1$ meet in point $C_2$. Prove that all the lines $C_1C_2$ have a common point.

2013 Sharygin Geometry Olympiad First Round p21 grades 10-11

Let $A$ be a point inside a circle $\omega$. One of two lines drawn through $A$ intersects $\omega$ at points $B$ and $C$, the second one intersects it at points $D$ and $E$ ($D$ lies between $A$ and $E$). The line passing through $D$ and parallel to $BC$ meets $\omega$ for the second time at point $F$, and the line $AF$ meets $\omega$ at point $T$. Let $M$ be the common point of the lines $ET$ and $BC$, and $N$ be the reflection of $A$ across $M$. Prove that the circumcircle of triangle $DEN$ passes through the midpoint of segment $BC$.

2013 Sharygin Geometry Olympiad First Round p22 grades 10-11

The common perpendiculars to the opposite sidelines of a nonplanar quadrilateral are mutually orthogonal. Prove that they intersect.

2013 Sharygin Geometry Olympiad First Round p23 grades 10-11

Two convex polytopes $A$ and $B$ do not intersect. The polytope $A$ has exactly $2012$ planes of symmetry. What is the maximal number of symmetry planes of the union of $A$ and $B$, if $B$ has a) $2012$, 

b) $2013$ symmetry planes?

c) What is the answer to the question of p.b), if the symmetry planes are replaced by the symmetry axes?

2012-2013 Final Round

grade 8

2013 Sharygin Geometry Olympiad Finals 8.1

Let $ABCDE$ be a pentagon with right angles at vertices $B$ and $E$ and such that $AB = AE$ and $BC = CD = DE$. The diagonals $BD$ and $CE$ meet at point $F$. Prove that $FA = AB$.

2013 Sharygin Geometry Olympiad Finals 8.2

Two circles with centers $O_1$ and $O_2$ meet at points $A$ and $B$. The bisector of angle $O_1AO_2$ meets the circles for the second time at points $C$and $D$. Prove that the distances from the circumcenter of triangle $CBD$ to $O_1$ and to $O_2$ are equal.

2013 Sharygin Geometry Olympiad Finals 8.3

Each vertex of a convex polygon is projected to all nonadjacent sidelines. Can it happen that each of these projections lies outside the corresponding side?

2013 Sharygin Geometry Olympiad Finals 8.4

The diagonals of a convex quadrilateral $ABCD$ meet at point $L$. The orthocenter $H$ of the triangle $LAB$ and the circumcenters $O_1, O_2$, and $O_3$ of the triangles $LBC, LCD$, and $LDA$ were marked. Then the whole configuration except for points $H, O_1, O_2$, and $O_3$ was erased. Restore it using a compass and a ruler.

2013 Sharygin Geometry Olympiad Finals 8.5

The altitude $AA'$, the median $BB'$, and the angle bisector $CC'$ of a triangle $ABC$ are concurrent at point $K$. Given that $A'K = B'K$, prove that $C'K = A'K$.

2013 Sharygin Geometry Olympiad Finals 8.6

Let $\alpha$ be an arc with endpoints $A$ and $B$ (see fig. ). A circle $\omega$ is tangent to segment $AB$ at point $T$ and meets $\alpha$  at points $C$ and $D$. The rays $AC$ and $TD$ meet at point $E$, while the rays $BD$ and $TC$ meet at point $F$. Prove that $EF$ and $AB$ are parallel.

2013 Sharygin Geometry Olympiad Finals 8.7

In the plane, four points are marked. It is known that these points are the centers of four circles, three of which are pairwise externally tangent, and all these three are internally tangent to the fourth one. It turns out, however, that it is impossible to determine which of the marked points is the center of the fourth (the largest) circle. Prove that these four points are the vertices of a rectangle.

2013 Sharygin Geometry Olympiad Finals 8.8

Let P be an arbitrary point on the arc $AC$ of the circumcircle of a fixed triangle $ABC$, not containing $B$. The bisector of angle $APB$ meets the bisector of angle $BAC$ at point $P_a$ the bisector of angle $CPB$ meets the bisector of angle $BCA$ at point $P_c$. Prove that for all points $P$, the circumcenters of triangles $PP_aP_c$ are collinear.

grade 9

2013 Sharygin Geometry Olympiad Finals 9.1

All angles of a cyclic pentagon $ABCDE$ are obtuse. The sidelines $AB$ and $CD$ meet at point $E_1$, the sidelines $BC$ and $DE$ meet at point $A_1$. The tangent at $B$ to the circumcircle of the triangle $BE_1C$ meets the circumcircle $\omega$ of the pentagon for the second time at point $B_1$. The tangent at $D$ to the circumcircle of the triangle $DA_1C$ meets  $\omega$  for the second time at point $D_1$. Prove that $B_1D_1 // AE$

2013 Sharygin Geometry Olympiad Finals 9.2

Two circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ meet at points $A$ and $B$. Points $C$ and $D$ on $\omega_1$ and $\omega_2$, respectively, lie on the opposite sides of the line $AB$ and are equidistant from this line. Prove that $C$ and $D$ are equidistant from the midpoint of $O_1O_2$.

2013 Sharygin Geometry Olympiad Finals 9.3

Each sidelength of a convex quadrilateral $ABCD$ is not less than $1$ and not greater than $2$. The diagonals of this quadrilateral meet at point $O$. Prove that $S_{AOB}+ S_{COD} \le  2(S_{AOD}+ S_{BOC})$.

2013 Sharygin Geometry Olympiad Finals 9.4

A point $F$ inside a triangle $ABC$ is chosen so that $\angle AFB = \angle BFC = \angle CFA$. The line passing through $F$ and perpendicular to $BC$ meets the median from $A$ at point $A_1$. Points $B_1$ and $C_1$ are defined similarly. Prove that the points $A_1, B_1$, and $C_1$ are three vertices of some regular hexagon, and that the three remaining vertices of that hexagon lie on the sidelines of $ABC$.

2013 Sharygin Geometry Olympiad Finals 9.5

Points $E$ and $F$ lie on the sides $AB$ and $AC$ of a triangle $ABC$. Lines $EF$ and $BC$ meet at point $S$. Let $M$ and $N$ be the midpoints of $BC$ and $EF$, respectively. The line passing through $A$ and parallel to $MN$ meets $BC$ at point $K$. Prove that $\frac{BK}{CK}=\frac{FS}{ES}$ .
.

2013 Sharygin Geometry Olympiad Finals 9.6 

A line $\ell$ passes through the vertex $B$ of a regular triangle $ABC$. A circle $\omega_a$ centered at $I_a$ is tangent to $BC$ at point $A_1$, and is also tangent to the lines $\ell$  and $AC$. A circle $\omega_c$ centered at $I_c$ is tangent to $BA$ at point $C_1$, and is also tangent to the lines $\ell$  and $AC$.  Prove that the orthocenter of triangle $A_1BC_1$ lies on the line $I_aI_c$.

2013 Sharygin Geometry Olympiad Finals 9.7

Two fixed circles $\omega_1$ and $\omega_2$ pass through point $O$. A circle of an arbitrary radius $R$ centered at $O$ meets $\omega_1$ at points $A$ and $B$, and meets $\omega_2$ at points $C$ and $D$. Let $X$ be the common point of lines $AC$ and $BD$. Prove that all the points X are collinear as $R$ changes.

2013 Sharygin Geometry Olympiad Finals 9.8

Three cyclists ride along a circular road with radius $1$ km counterclockwise. Their velocities are constant and different. Does there necessarily exist (in a sufficiently long time) a moment when all the three distances between cyclists are greater than $1$ km?

grade 10

2013 Sharygin Geometry Olympiad Finals 10.1

A circle $k$ passes through the vertices $B$ and $C$ of a triangle $ABC$ with $AB > AC$. This circle meets the extensions of sides $AB$ and $AC$ beyond $B$ and $C$ at points $P$ and $Q$, respectively. Let $AA_1$ be the altitude of $ABC$. Given that $A_1P = A_1Q$, prove that $\angle PA_1Q = 2\angle BAC$.

2013 Sharygin Geometry Olympiad Finals 10.2

Let  $ABCD$ be a circumscribed quadrilateral with $AB = CD \ne BC$. The diagonals of the quadrilateral meet at point $L$. Prove that the angle $ALB$ is acute.

2013 Sharygin Geometry Olympiad Finals 10.3

Let $X$ be a point inside a triangle $ABC$ such that $XA \cdot  BC = XB \cdot  AC = XC \cdot  AB$. Let $I_1, I_2$, and $I_3$ be the incenters of the triangles $XBC, XCA$, and $XAB$, respectively. Prove that the lines $AI_1, BI_2$, and $CI_3$ are concurrent.

2013 Sharygin Geometry Olympiad Finals 10.4

We are given a cardboard square of area $1/4$ and a paper triangle of area $1/2$ such that all the squares of the side lengths of the triangle are integers. Prove that the square can be completely wrapped with the triangle. (In other words, prove that the triangle can be folded along several straight lines and the square can be placed inside the folded figure so that both faces of the square are completely covered with paper.)

2013 Sharygin Geometry Olympiad Finals 10.5 

Let $O$ be the circumcenter of a cyclic quadrilateral $ABCD$. Points $E$ and $F$ are the midpoints of arcs $AB$ and $CD$ not containing the other vertices of the quadrilateral. The lines passing through $E$ and $F$ and parallel to the diagonals of $ABCD$ meet at points $E, F, K$, and $L$. Prove that line $KL$ passes through $O$.

2013 Sharygin Geometry Olympiad Finals 10.6

The altitudes $AA_1, BB_1$, and $CC_1$ of an acute-angled triangle $ABC$ meet at point $H$. The perpendiculars from $H$ to $B_1C_1$ and $A_1C_1$ meet the rays $CA$ and $CB$ at points $P$ and $Q$, respectively. Prove that the perpendicular from $C$ to $A_1B_1$ passes through the midpoint of $PQ$.

2013 Sharygin Geometry Olympiad Finals 10.7

In the space, five points are marked. It is known that these points are the centers of five spheres, four of which are pairwise externally tangent, and all these four are internally tangent to the fifth one. It turns out, however, that it is impossible to determine which of the marked points is the center of the fifth (the largest) sphere. Find the ratio of the greatest and the smallest radii of the spheres.

2013 Sharygin Geometry Olympiad Finals 10.8

In the plane, two fixed circles are given, one of them lies inside the other one. For an arbitrary point $C$ of the external circle, let $CA$ and $CB$ be two chords of this circle which are tangent to the internal one. Find the locus of the incenters of triangles $ABC$.

2013-2014 First Round

2014 Sharygin Geometry Olympiad First Round p1 grade 8

A right-angled triangle $ABC$ is given. Its catheus $AB$ is the base of a regular triangle $ADB$ lying in the exterior of $ABC$, and its hypotenuse $AC$ is the base of a regular triangle $AEC$ lying in the interior of $ABC$. Lines $DE$ and $AB$ meet at point $M$. The whole configuration except points $A$ and $B$ was erased. Restore the point $M$.

2014 Sharygin Geometry Olympiad First Round p2 grade 8

A paper square with sidelength $2$ is given. From this square, can we cut out a $12$-gon having all sidelengths equal to $1$ and all angles divisible by $45^\circ$?

2014 Sharygin Geometry Olympiad First Round p3 grade 8

Let $ABC$ be an isosceles triangle with base $AB$. Line $\ell$ touches its circumcircle at point $B$. Let $CD$ be a perpendicular from $C$ to $\ell$, and $AE$, $BF$ be the altitudes of $ABC$. Prove that $D$, $E$, and $F$ are collinear.

2014 Sharygin Geometry Olympiad First Round p4 grade 8

A square is inscribed into a triangle (one side of the triangle contains two vertices and each of two remaining sides contains one vertex. Prove that the incenter of the triangle lies inside the square.

2014 Sharygin Geometry Olympiad First Round p5 grade 8

In an acute-angled triangle $ABC$, $AM$ is a median, $AL$ is a bisector and $AH$ is an altitude ($H$ lies between $L$ and $B$). It is known that $ML=LH=HB$. Find the ratios of the sidelengths of $ABC$.

2014 Sharygin Geometry Olympiad First Round p6 grades 8-9

Given a circle with center $O$ and a point $P$ not lying on it, let $X$ be an arbitrary point on this circle and $Y$ be a common point of the bisector of angle $POX$ and the perpendicular bisector to segment $PX$. Find the locus of points $Y$.

2014 Sharygin Geometry Olympiad First Round p7 grades 8-9

A parallelogram $ABCD$ is given. The perpendicular from $C$ to $CD$ meets the perpendicular from $A$ to $BD$ at point $F$, and the perpendicular from $B$ to $AB$ meets the perpendicular bisector to $AC$ at point $E$. Find the ratio in which side $BC$ divides segment $EF$.

2014 Sharygin Geometry Olympiad First Round p8 grades 8-9

Let $ABCD$ be a rectangle. Two perpendicular lines pass through point $B$. One of them meets segment $AD$ at point $K$, and the second one meets the extension of side $CD$ at point $L$. Let $F$ be the common point of $KL$ and $AC$. Prove that $BF\perp KL$.

2014 Sharygin Geometry Olympiad First Round p9 grades 8-9

Two circles $\omega_1$ and $\omega_2$ touching externally at point $L$ are inscribed into angle $BAC$. Circle $\omega_1$ touches ray $AB$ at point $E$, and circle $\omega_2$ touches ray $AC$ at point $M$. Line $EL$ meets $\omega_2$ for the second time at point $Q$. Prove that $MQ\parallel AL$.

2014 Sharygin Geometry Olympiad First Round p10 grades 8-9
Two disjoint circles $\omega_1$ and $\omega_2$ are inscribed into an angle. Consider all pairs of parallel lines $l_1$ and $l_2$ such that $l_1$ touches $\omega_1$ and $l_2$ touches $\omega_2$ ($\omega_1$, $\omega_2$ lie between $l_1$ and $l_2$). Prove that the medial lines of all trapezoids formed by $l_1$ and $l_2$ and the sides of the angle touch some fixed circle.

2014 Sharygin Geometry Olympiad First Round p11 grades 8-9

Points $K, L, M$ and $N$ lying on the sides $AB, BC, CD$ and $DA$ of a square $ABCD$ are vertices of another square. Lines $DK$ and $N M$ meet at point $E$, and lines $KC$ and $LM$ meet at point $F$ . Prove that $EF\parallel AB$.

2014 Sharygin Geometry Olympiad First Round p12 grades 9-10

Circles $\omega_1$ and $\omega_2$ meet at points $A$ and $B$. Let points $K_1$ and $K_2$ of $\omega_1$ and $\omega_2$ respectively be such that $K_1A$ touches $\omega_2$, and $K_2A$ touches $\omega_1$. The circumcircle of triangle $K_1BK_2$ meets lines $AK_1$ and $AK_2$ for the second time at points $L_1$ and $L_2$ respectively. Prove that $L_1$ and $L_2$ are equidistant from line $AB$.

2014 Sharygin Geometry Olympiad First Round p13 grades 9-10

Let $AC$ be a fixed chord of a circle $\omega$ with center $O$. Point $B$ moves along the arc $AC$. A fixed point $P$ lies on $AC$. The line passing through $P$ and parallel to $AO$ meets $BA$ at point $A_1$, the line passing through $P$ and parallel to $CO$ meets $BC$ at point $C_1$. Prove that the circumcenter of triangle $A_1BC_1$ moves along a straight line.

2014 Sharygin Geometry Olympiad First Round p14 grades 9-11

In a given disc, construct a subset such that its area equals the half of the disc area and its intersection with its reflection over an arbitrary diameter has the area equal to the quarter of the disc area.

2014 Sharygin Geometry Olympiad First Round p15 grades 9-11

Let $ABC$ be a non-isosceles triangle. The altitude from $A$, the bisector from $B$ and the median from $C$ concur at point $K$.
a) Which of the sidelengths of the triangle is medial (intermediate in length)?
b) Which of the lengths of segments $AK, BK, CK$ is medial (intermediate in length)?

2014 Sharygin Geometry Olympiad First Round p16 grades 9-11

Given a triangle $ABC$ and an arbitrary point $D$.The lines passing through $D$ and perpendicular to segments $DA, DB, DC$ meet lines $BC, AC, AB$ at points $A_1, B_1, C_1$ respectively. Prove that the midpoints of segments $AA_1, BB_1, CC_1$ are collinear.

2014 Sharygin Geometry Olympiad First Round p17 grades 10-11

Let $AC$ be the hypothenuse of a right-angled triangle $ABC$. The bisector $BD$ is given, and the midpoints $E$ and $F$ of the arcs $BD$ of the circumcircles of triangles $ADB$ and $CDB$ respectively are marked (the circles are erased). Construct the centers of these circles using only a ruler.

2014 Sharygin Geometry Olympiad First Round p18 grades 10-11

Let $I$ be the incenter of a circumscribed quadrilateral $ABCD$. The tangents to circle $AIC$ at points $A, C$ meet at point $X$. The tangents to circle $BID$ at points $B, D$ meet at point $Y$ . Prove that $X, I, Y$ are collinear.

2014 Sharygin Geometry Olympiad First Round p19 grades 10-11

Two circles $\omega_1$ and $\omega_2$ touch externally at point $P$.Let $A$ be a point on $\omega_2$ not lying on the line through the centres of the two circles.Let $AB$ and $AC$ be the tangents to $\omega_1$.Lines $BP$ and $CP$ meet  $\omega_2$ for the second time at points $E$ and $F$.Prove that the line $EF$,the tangent to $\omega_2$ at $A$ and the common tangent at $P$ concur.

2014 Sharygin Geometry Olympiad First Round p20 grades 10-11

A quadrilateral $KLMN$ is given. A circle with center $O$ meets its side $KL$ at points $A$ and $A_1$, side $LM$ at points $B$ and $B_1$, etc. Prove that if the circumcircles of triangles $KDA, LAB, MBC$ and $NCD$ concur at point $P$, then
a) the circumcircles of triangles $KD_1A_1, LA_1B_1, MB_1C_1$ and $NC1D1$ also concur at some point $Q$;
b) point $O$ lies on the perpendicular bisector to $PQ$.

2014 Sharygin Geometry Olympiad First Round p21 grades 10-11

Let $ABCD$ be a circumscribed quadrilateral. Its incircle $\omega$ touches the sides $BC$ and $DA$ at points $E$ and $F$ respectively. It is known that lines $AB,FE$ and  $CD$ concur. The circumcircles of triangles $AED$ and $BFC$ meet $\omega$ for the second time at points $E_1$ and $F_1$. Prove that $EF$ is parallel to $E_1 F_1$.

2014 Sharygin Geometry Olympiad First Round p22 grades 10-11

Does there exist a convex polyhedron such that it has diagonals and each of them is shorter than each of its edges?

2014 Sharygin Geometry Olympiad First Round p23 grade 11

Let $A, B, C$ and $D$ be a triharmonic quadruple of points, i.e $AB\cdot CD = AC \cdot BD = AD \cdot BC.$
Let $A_1$ be a point distinct from $A$ such that the quadruple $A_1, B, C$ and $D$ is triharmonic.
Points $B_1, C_1$ and $D_1$ are defined similarly. Prove that
a) $A, B, C_1, D_1$ are concyclic;
b) the quadruple $A_1, B_1, C_1, D_1$ is triharmonic.

2014 Sharygin Geometry Olympiad First Round p24 grade 11

A circumscribed pyramid $ABCDS$ is given. The opposite sidelines of its base meet at points $P$ and $Q$ in such a way that $A$ and $B$ lie on segments $PD$ and $PC$ respectively. The inscribed sphere touches faces $ABS$ and $BCS$ at points $K$ and $L$. Prove that if $PK$ and $QL$ are complanar then the touching point of the sphere with the base lies on $BD$.

2013-2014 Final Round

grade 8

2014 Sharygin Geometry Olympiad Finals 8.1

The incircle of a right-angled triangle $ABC$ touches its catheti $AC$ and $BC$ at points $B_1$ and $A_1$, the hypotenuse touches the incircle at point $C_1$. Lines $C_1A_1$ and $C_1B_1$ meet $CA$ and $CB$ respectively at points $B_0$ and $A_0$. Prove that $AB_0 = BA_0$.

by J. Zajtseva, D. Shvetsov

2014 Sharygin Geometry Olympiad Finals 8.2 

Let $AH_a$ and $BH_b$ be altitudes, $AL_a$ and $BL_b$ be angle bisectors of a triangle $ABC$. It is known that $H_aH_b // L_aL_b$. Is it necessarily true that $AC = BC$?

by B. Frenkin

2014 Sharygin Geometry Olympiad Finals 8.3

Points $M$ and $N$ are the midpoints of sides $AC$ and $BC$ of a triangle $ABC$. It is known that $\angle MAN = 15 ^ o$ and $\angle BAN = 45 ^ o$. Find the value of angle $\angle ABM$.

by A. Blinkov

2014 Sharygin Geometry Olympiad Finals 8.4

Tanya has cut out a triangle from checkered paper as shown in the picture. The lines of the grid have faded. Can Tanya restore them without any instruments only folding the triangle (she remembers the triangle sidelengths)?

by T. Kazitsyna

2014 Sharygin Geometry Olympiad Finals 8.5

A triangle with angles of $30, 70$ and $80$ degrees is given. Cut it by a straight line into two triangles in such a way that an angle bisector in one of these triangles and a median in the other one drawn from two endpoints of the cutting segment are parallel to each other. (It suffices to find one such cutting.)

by A. Shapovalov

2014 Sharygin Geometry Olympiad Finals 8.6

Two circles $k_1$ and $k_2$ with centers $O_1$ and $O_2$ are tangent to each other externally at point $O$. Points $X$ and $Y$ on $k_1$ and $k_2$ respectively are such that rays $O_1X$ and $O_2Y$ are parallel and codirectional. Prove that two tangents from $X$ to $k_2$ and two tangents from $Y$ to $k_1$ touch the same circle passing through $O$.

by V. Yasinsky

2014 Sharygin Geometry Olympiad Finals 8.7

Two points on a circle are joined by a broken line shorter than the diameter of the circle. Prove that there exists a diameter which does not intersect this broken line.

Folklore

2014 Sharygin Geometry Olympiad Finals 8.8

Let $M$ be the midpoint of the chord $AB$ of a circle centered at $O$. Point $K$ is symmetric to $M$ with respect to $O$, and point $P$ is chosen arbitrarily on the circle. Let $Q$ be the intersection of the line perpendicular to $AB$ through $A$ and the line perpendicular to $PK$ through $P$. Let $H$ be the projection of $P$ onto $AB$. Prove that $QB$ bisects $PH$.

by Tran Quang Hung

grade 9

2014 Sharygin Geometry Olympiad Finals 9.1

Let ABCD be a quadrilateral cyclic. Prove that $AC> BD$ if and only if $(AD-BC) (AB-CD)> 0$.

by V. Yasinsky

2014 Sharygin Geometry Olympiad Finals 9.2

In a quadrilateral $ABCD$ angles $A$ and $C$ are right. Two circles with diameters $AB$ and $CD$ meet at points $X$ and $Y$. Prove that line $XY$ passes through the midpoint of $AC$.

by F. Nilov

2014 Sharygin Geometry Olympiad Finals 9.p3

An acute angle $A$ and a point $E$ inside it are given. Construct points $B, C$ on the sides of the angle such that $E$ is the center of the Euler circle of triangle $ABC$.

by E. Diomidov

2014 Sharygin Geometry Olympiad Finals 9.4

Let $H$ be the orthocenter of a triangle $ABC$. Given that $H$ lies on the incircle of $ABC$, prove that three circles with centers $A, B, C$ and radii $AH, BH, CH$ have a common tangent.

by Mahdi Etesami Fard

2014 Sharygin Geometry Olympiad Finals 9.5

In triangle $ABC$ $\angle B = 60 ^ o, O$ is the circumcenter, and $L$ is the foot of an angle bisector of angle $B$. The circumcirle of triangle $BOL$ meets the circumcircle of $ABC$ at point $D \ ne B$. Prove that $BD \ perp AC$.

by D. Shvetsov

2014 Sharygin Geometry Olympiad Finals 9.6

Let $I$ be the incenter of triangle $ABC$, and $M, N$ be the midpoints of arcs $ABC$ and $BAC$ of its circumcircle. Prove that points $M, I, N$ are collinear if and only if $AC + BC = 3AB$.

by A. Polyansky

2014 Sharygin Geometry Olympiad Final s 9.7

Nine circles are drawn around an arbitrary triangle as in the figure. All circles tangent to the same side of the triangle have equal radii. Three lines are drawn, each one connecting one of the triangle's vertices to the center of one of the circles touching the opposite side, as in the figure. Show that the three lines are concurrent.

by N. Beluhov

2014 Sharygin Geometry Olympiad Finals 9.8

A convex polygon $P$ lies on a flat wooden table. You are allowed to drive some nails into the table. The nails must not go through $P$, but they may touch its boundary. We say that a set of nails blocks $P$ if the nails make it impossible to move $P$ without lifting it off the table. What is the minimum number of nails that suffices to block any convex polygon $P$?

by N. Beluhov, S. Gerdgikov

grade 10

2014 Sharygin Geometry Olympiad Finals 10.1

The vertices and the circumcenter of an isosceles triangle lie on four different sides of a square. Find the angles of this triangle.

by I. Bogdanov, B. Frenkin

2014 Sharygin Geometry Olympiad Finals 10.2 

A circle, its chord $AB$ and the midpoint $W$ of the minor arc $AB$ are given. Take an arbitrary point $C$ on the major arc $AB$. The tangent to the circle at $C$ meets the tangents at $A$ and $B$ at points $X$ and $Y$ respectively. Lines $WX$ and WY meet AB at points $N$ and $M$ respectively. Prove that the length of segment $NM$ does not depend on point $C$.

by A. Zertsalov, D. Skrobot

2014 Sharygin Geometry Olympiad Finals 10.3

Do there exist convex polyhedra with an arbitrary number of diagonals (a diagonal is a segment joining two vertices of a polyhedron and not lying on the surface of this polyhedron)?

by A. Blinkov

2014 Sharygin Geometry Olympiad Finals 10.4

Let $ABC$ be a fixed triangle in the plane. Let $D$ be an arbitrary point in the plane. The circle with center $D$, passing through $A$, meets $AB$ and $AC$ again at points $A_b$ and $A_c$ respectively. Points $B_a, B_c, C_a$ and $C_b$ are defined similarly. A point $D$ is called good if the points $A_b, A_c, B_a, B_c, C_a$, and $C_b$ are concyclic. For a given triangle $ABC$, how many good points can there be?

by A. Garkavyj, A. Sokolov

2014 Sharygin Geometry Olympiad Finals 10.5

The altitude from one vertex of a triangle, the bisector from the other one and the median from the remaining vertex were drawn, the common points of these three lines were marked, and after this everything was erased except three marked points. Restore the triangle. (For every two erased segments, it is known which of the three points was their intersection point.)

by A. Zaslavsky

2014 Sharygin Geometry Olympiad Finals 10.66

The incircle of a non-isosceles triangle $ABC$ touches $AB$ at point $C '$. The circle with diameter $BC '$ meets the incircle and the bisector of angle $B$ again at points $A_1$ and $A_2$ respectively. The circle with diameter $AC '$ meets the incircle and the bisector of angle $A$ again at points $B_1$ and $B_2$ respectively. Prove that lines $AB, A_1B_1, A_2B_2$ concur.

by EH Garsia

2014 Sharygin Geometry Olympiad Finals 10.7

Prove that the smallest dihedral angle between faces of an arbitrary tetrahedron is not greater than the dihedral angle between faces of a regular tetrahedron.

by S. Shosman, O. Ogievetsky

2014 Sharygin Geometry Olympiad Finals 10.8

Given is a cyclic quadrilateral $ABCD$. The point $L_a$ lies in the interior of $BCD$ and is such that its distances to the sides of this triangle are proportional to the lengths of corresponding sides. The points $L_b, L_c$, and $L_d$ are defined analogously. Given that the quadrilateral $L_aL_bL_cL_d$ is cyclic, prove that the quadrilateral $ABCD$ has two parallel sides.

 by N. Beluhov

2014-2015 First Round

2015 Sharygin Geometry Olympiad First Round p1 grade 8

Tanya cut out a convex polygon from the paper, fold it several times and obtained a two-layers quadrilateral. Can the cutted polygon be a heptagon?

2015 Sharygin Geometry Olympiad First Round p2 grade 8

Let $O$ and $H$ be the circumcenter and the orthocenter of a triangle $ABC$. The line passing through the midpoint of $OH$ and parallel to $BC$ meets $AB$ and $AC$ at points $D$ and $E$. It is known that $O$ is the incenter of triangle $ADE$. Find the angles of $ABC$.

2015 Sharygin Geometry Olympiad First Round p3 grade 8

The side $AD$ of a square $ABCD$ is the base of an obtuse-angled isosceles triangle $AED$ with vertex $E$ lying inside the square. Let $AF$ be a diameter of the circumcircle of this triangle, and $G$ be a point on $CD$ such that $CG = DF$. Prove that angle $BGE$ is less than half of angle $AED$.

2015 Sharygin Geometry Olympiad First Round p4 grade 8

In a parallelogram $ABCD$ the trisectors of angles $A$ and $B$ are drawn. Let $O$ be the common points of the trisectors nearest to $AB$. Let $AO$ meet the second trisector of angle $B$ at point $A_1$, and let $BO$ meet the second trisector of angle $A$ at point $B_1$. Let $M$ be the midpoint of $A_1B_1$. Line $MO$ meets $AB$ at point $N$ Prove that triangle $A_1B_1N$ is equilateral.

2015 Sharygin Geometry Olympiad First Round p5 grades 8-9

Let a triangle $ABC$ be given. Two circles passing through $A$ touch $BC$ at points $B$ and $C$ respectively. Let $D$ be the second common point of these circles  ($A$ is closer to $BC$ than $D$). It is known that $BC = 2BD$. Prove that $\angle DAB = 2\angle ADB.$

2015 Sharygin Geometry Olympiad First Round p6 grades 8-9

Let $AA', BB'$ and $CC'$ be the altitudes of an acute-angled triangle $ABC$. Points $C_a, C_b$ are symmetric to $C'$ wrt $AA'$ and $BB'$. Points $A_b, A_c, B_c, B_a$ are defined similarly. Prove that lines $A_bB_a, B_cC_b$ and $C_aA_c$ are parallel.

2015 Sharygin Geometry Olympiad First Round p7 grades 8-9

The altitudes $AA_1$ and $CC_1$ of a triangle $ABC$ meet at point $H$. Point $H_A$ is symmetric to $H$ about $A$. Line $H_AC_1$ meets $BC$ at point $C'$, point $A'$ is defined similarly. Prove that $A' C' // AC$.

2015 Sharygin Geometry Olympiad First Round p8 grades 8-9

Diagonals of an isosceles trapezoid $ABCD$ with bases $BC$ and $AD$ are perpendicular. Let $DE$ be the perpendicular from $D$ to $AB$, and let $CF$ be the perpendicular from $C$ to $DE$. Prove that angle $DBF$ is equal to half of angle $FCD$.

2015 Sharygin Geometry Olympiad First Round p9 grades 8-9

Let $ABC$ be an acute-angled triangle. Construct points $A', B', C'$ on its sides $BC, CA, AB$ such that:
- $A'B'  // AB$,
- $C'C$ is the bisector of angle $A'C'B'$,
- $A'C' + B'C'= AB$.

2015 Sharygin Geometry Olympiad First Round p10 grades 8-9

The diagonals of a convex quadrilateral divide it into four similar triangles. Prove that is possible to inscribe a circle into this quadrilateral.

2015 Sharygin Geometry Olympiad First Round p11 grades 8-10

Let $H$ be the orthocenter of an acute-angled triangle A$BC$. The perpendicular bisector to segment $BH$ meets $BA$ and $BC$ at points $A_0, C_0$ respectively. Prove that the perimeter of triangle $A_0OC_0$ ($O$ is the circumcenter of \triangle $ABC$) is equal to $AC$.

2015 Sharygin Geometry Olympiad First Round p12 grades 8-11

Find the maximal number of discs which can be disposed on the plane so that each two of them have a common point and no three have it

2015 Sharygin Geometry Olympiad First Round p13 grades 9-10

Let $AH_1, BH_2$ and $CH_3$ be the altitudes of a triangle $ABC$. Point $M$ is the midpoint of $H_2H_3$. Line $AM$ meets $H_2H_1$ at point $K$. Prove that $K$ lies on the medial line of $ABC$ parallel to $AC$.

2015 Sharygin Geometry Olympiad First Round p14 grades 9-11

Let $ABC$ be an acute-angled, nonisosceles triangle. Point $A_1, A_2$ are symmetric to the feet of the internal and the external bisectors of angle $A$ wrt the midpoint of $BC$. Segment $A_1A_2$ is a diameter of a circle $\alpha$. Circles $\beta$ and $\gamma$ are defined similarly. Prove that these three circles have two common points

2015 Sharygin Geometry Olympiad First Round p15 grades 9-11

The sidelengths of a triangle $ABC$ are not greater than $1$. Prove that $p(1 -2Rr)$ is not greater than $1$, where $p$ is the semiperimeter, $R$ and $r$ are the circumradius and the inradius of $ABC$.

2015 Sharygin Geometry Olympiad First Round p16 grades 9-11

he diagonals of a convex quadrilateral divide it into four triangles. Restore the quadrilateral by the circumcenters of two adjacent triangles and the incenters of two mutually opposite triangles.

2015 Sharygin Geometry Olympiad First Round p17 grades 10-11

Let $O$ be the circumcenter of a triangle $ABC$. The projections of points $D$ and $X$ to the sidelines of the triangle lie on lines $\ell$ and $L$ such that $\ell // XO$. Prove that the angles formed by $L$ and by the diagonals of quadrilateral $ABCD$ are equal.

2015 Sharygin Geometry Olympiad First Round p18 grades 10-11

Let $ABCDEF$ be a cyclic hexagon, points $K, L, M, N$ be the common points of lines $AB$ and $CD$, $AC$ and $BD$, $AF$ and $DE$, $AE$ and $DF$ respectively. Prove that if three of these points are collinear then the fourth point lies on the same line.

2015 Sharygin Geometry Olympiad First Round p19 grades 10-11

Let $L$ and $K$ be the feet of the internal and the external bisector of angle $A$ of a triangle $ABC$. Let $P$ be the common point of the tangents to the circumcircle of the triangle at $B$ and $C$. The perpendicular from $L$ to $BC$ meets $AP$ at point $Q$. Prove that $Q$ lies on the medial line of triangle $LKP$.

2015 Sharygin Geometry Olympiad First Round p20 grades 10-11

Let $L$ and $K$ be the feet of the internal and the external bisector of angle $A$ of a triangle $ABC$. Let $P$ be the common point of the tangents to the circumcircle of the triangle at $B$ and $C$. The perpendicular from $L$ to $BC$ meets $AP$ at point $Q$. Prove that $Q$ lies on the medial line of triangle $LKP$.

2015 Sharygin Geometry Olympiad First Round p21 grades 10-11

Let $L$ and $K$ be the feet of the internal and the external bisector of angle $A$ of a triangle $ABC$. Let $P$ be the common point of the tangents to the circumcircle of the triangle at $B$ and $C$. The perpendicular from $L$ to $BC$ meets $AP$ at point $Q$. Prove that $Q$ lies on the medial line of triangle $LKP$.

2015 Sharygin Geometry Olympiad First Round p22 grades 10-11

The faces of an icosahedron are painted into $5$ colors in such a way that two faces painted into the same color have no common points, even a vertices. Prove that for any point lying inside the icosahedron the sums of the distances from this point to the red faces and the blue faces are equal.

2015 Sharygin Geometry Olympiad First Round p23 grade 11

A tetrahedron $ABCD$ is given. The incircles of triangles $ABC$ and $ABD$ with centers $O_1, O_2$, touch $AB$ at points $T_1, T_2$. The plane $\pi_{AB}$ passing through the midpoint of $T_1T_2$ is perpendicular to $O_1O_2$. The planes $\pi_{AC},\pi_{BC}, \pi_{AD}, \pi_{BD}, \pi_{CD}$ are defined similarly. Prove that these six planes have a common point.

2015 Sharygin Geometry Olympiad First Round p24 grade 11

The insphere of a tetrahedron ABCD with center $O$ touches its faces at points $A_1,B_1,C_1$ and  $D_1$.

a) Let $P_a$ be a point such that its reflections in lines $OB,OC$ and $OD$ lie on plane $BCD$.

Points $P_b, P_c$ and $P_d$ are defined similarly. Prove that lines $A_1P_a,B_1P_b,C_1P_c$ and $D_1P_d$ concur at some point $P$.

b) Let $I$ be the incenter of $A_1B_1C_1D_1$ and $A_2$ the common point of line $A_1I$ with plane $B_1C_1D_1$. Points $B_2, C_2, D_2$ are defined similarly. Prove that $P$ lies inside $A_2B_2C_2D_2$.

2014 -2015 Final Round

grade 8

2015 Sharygin Geometry Olympiad Finals 8.1

In trapezoid $ABCD$ angles $A$ and $B$ are right, $AB = AD, CD = BC + AD, BC < AD$. Prove that $\angle ADC = 2\angle ABE$, where $E$ is the midpoint of segment $AD$.

by V. Yasinsky

2015 Sharygin Geometry Olympiad Finals 8.2 

A circle passing through $A, B$ and the orthocenter of triangle $ABC$ meets sides $AC, BC$ at their inner points. Prove that $60^o < \angle C < 90^o$ .

by A. Blinkov

2015 Sharygin Geometry Olympiad Finals 8.3

In triangle $ABC$ we have $AB = BC, \angle B = 20^o$. Point $M$ on $AC$ is such that $AM : MC = 1 : 2$, point $H$ is the projection of $C$ to $BM$. Find angle $AHB$.

by M. Yevdokimov

2015 Sharygin Geometry Olympiad Finals 8.4 

Prove that an arbitrary convex quadrilateral can be divided into five polygons having symmetry axes.

by N. Belukhov

2015 Sharygin Geometry Olympiad Finals 8.5

Two equal hard triangles are given. One of their angles is equal to $\alpha$ (these angles are marked). Dispose these triangles on the plane in such a way that the angle formed by some three vertices would be equal to $\alpha / 2$.
(No instruments are allowed, even a pencil.)

by E. Bakayev, A. Zaslavsky

2015 Sharygin Geometry Olympiad Finals 8.6 

Lines $b$ and $c$ passing through vertices $B$ and $C$ of triangle $ABC$ are perpendicular to sideline $BC$. The perpendicular bisectors to $AC$ and $AB$ meet $b$ and $c$ at points $P$ and $Q$ respectively. Prove that line $PQ$ is perpendicular to median $AM$ of triangle $ABC$.

by D. Prokopenko

2015 Sharygin Geometry Olympiad Finals8.7 

Point $M$ on side $AB$ of quadrilateral $ABCD$ is such that quadrilaterals $AMCD$ and $BMDC$ are circumscribed around circles centered at $O_1$ and $O_2$ respectively. Line $O_1O_2$ cuts an isosceles triangle with vertex M from angle $CMD$. Prove that $ABCD$ is a cyclic quadrilateral.

by M. Kungozhin

2015 Sharygin Geometry Olympiad Finals 8.8

Points $C_1, B_1$ on sides $AB, AC$ respectively of triangle $ABC$ are such that $BB_1 \perp  CC_1$. Point $X$ lying inside the triangle is such that $\angle XBC = \angle B_1BA, \angle XCB = \angle C_1CA$. Prove that $\angle B_1XC_1 =90^o- \angle A$.

by A. Antropov, A. Yakubov

grade 9

2015 Sharygin Geometry Olympiad Finals 9.1 

Circles $\alpha$ and  $\beta$ pass through point $C$. The tangent to $\alpha$ at this point meets $\beta$ at point $B$, and the tangent to  $\beta$ at $C$ meets $\alpha$ at point $A$ so that $A$ and $B$ are distinct from $C$ and angle $ACB$ is obtuse. Line $AB$ meets $\alpha$ and $\beta$ for the second time at points $N$ and $M$ respectively. Prove that $2MN < AB$.

by D. Mukhin

2015 Sharygin Geometry Olympiad Finals 9.2

A convex quadrilateral is given. Using a compass and a ruler construct a point such that its projections to the sidelines of this quadrilateral are the vertices of a parallelogram.

by A. Zaslavsky

2015 Sharygin Geometry Olympiad Finals 9.3 

Let $100$ discs lie on the plane in such a way that each two of them have a common point. Prove that there exists a point lying inside at least $15$ of these discs.

by M. Kharitonov, A. Polyansky

2015 Sharygin Geometry Olympiad Finals 9.4 

A fixed triangle $ABC$ is given. Point $P$ moves on its circumcircle so that segments $BC$ and $AP$ intersect. Line $AP$ divides triangle $BPC$ into two triangles with incenters $I_1$ and $I_2$. Line $I_1I_2$ meets $BC$ at point $Z$. Prove that all lines $ZP$ pass through a fixed point.

by R. Krutovsky, A. Yakubov

2015 Sharygin Geometry Olympiad Finals 9.5 

Let $BM$ be a median of nonisosceles right-angled triangle $ABC$ ($\angle B = 90^o$), and $Ha, Hc$ be the orthocenters of triangles $ABM, CBM$ respectively. Prove that lines $AH_c$ and $CH_a$ meet on the medial line of triangle $ABC$.

by D. Svhetsov

2015 Sharygin Geometry Olympiad Finals 9.6 

The diagonals of convex quadrilateral $ABCD$ are perpendicular. Points $A' , B' , C' , D'$ are the circumcenters of triangles $ABD, BCA, CDB, DAC$ respectively. Prove that lines $AA' , BB' , CC' , DD'$ concur.

by A. Zaslavsky

2015 Sharygin Geometry Olympiad Finals 9.7 

Let $ABC$ be an acute-angled, nonisosceles triangle. Altitudes $AA'$ and $BB'$meet at point $H$, and the medians of triangle $AHB$ meet at point $M$. Line $CM$ bisects segment $A'B'$. Find angle $C$.

by D. Krekov

2015 Sharygin Geometry Olympiad Finals 9.8 

A perpendicular bisector of side $BC$ of triangle $ABC$ meets lines $AB$ and $AC$ at points $A_B$ and $A_C$ respectively. Let $O_a$ be the circumcenter of triangle $AA_BA_C$. Points $O_b$ and $O_c$ are defined similarly. Prove that the circumcircle of triangle $O_aO_bO_c$ touches the circumcircle of the original triangle.

grade 10

2015 Sharygin Geometry Olympiad Finals 10.1

Let $K$ be an arbitrary point on side $BC$ of triangle $ABC$, and $KN$ be a bisector of triangle $AKC$. Lines $BN$ and $AK$ meet at point $F$, and lines $CF$ and $AB$ meet at point $D$. Prove that $KD$ is a bisector of triangle $AKB$.

2015 Sharygin Geometry Olympiad Finals 10.2 

Prove that an arbitrary triangle with area $1$ can be covered by an isosceles triangle with area less than $\sqrt{2}$.

2015 Sharygin Geometry Olympiad Finals 10.3 

Let $A_1$, $B_1$ and $C_1$ be the midpoints of sides $BC$, $CA$ and $AB$ of triangle $ABC$, respectively. Points $B_2$ and $C_2$ are the midpoints of segments $BA_1$ and $CA_1$ respectively. Point $B_3$ is symmetric to $C_1$ wrt $B$, and $C_3$ is symmetric to $B_1$ wrt $C$. Prove that one of common points of circles $BB_2B_3$ and $CC_2C_3$ lies on the circumcircle of triangle $ABC$.

2015 Sharygin Geometry Olympiad Finals 10.4 

Let $AA_1$, $BB_1$, $CC_1$ be the altitudes of an acute-angled, nonisosceles triangle $ABC$, and $A_2$, $B_2$, $C_2$ be the touching points of sides $BC$, $CA$, $AB$ with the correspondent excircles. It is known that line $B_1C_1$ touches the incircle of $ABC$. Prove that $A_1$ lies on the circumcircle of $A_2B_2C_2$.

2015 Sharygin Geometry Olympiad Finals 10.5 

Let $BM$ be a median of right-angled nonisosceles triangle $ABC$ ($\angle B = 90$), and $H_a$, $H_c$ be the orthocenters of triangles $ABM$, $CBM$ respectively. Lines  $AH_c$ and $CH_a$ meet at point $K$. Prove that $\angle MBK = 90$.

2015 Sharygin Geometry Olympiad Finals 10.6 

Let $H$ and $O$ be the orthocenter and the circumcenter of triangle $ABC$. The circumcircle of triangle $AOH$ meets the perpendicular bisector of $BC$ at point $A_1 \neq O$. Points $B_1$ and $C_1$ are defined similarly. Prove that lines $AA_1$, $BB_1$ and $CC_1$ concur.

2015 Sharygin Geometry Olympiad Finals 10.7 

Let $SABCD$ be an inscribed pyramid, and $AA_1$, $BB_1$, $CC_1$, $DD_1$ be the perpendiculars from $A$, $B$, $C$, $D$ to lines $SC$, $SD$, $SA$, $SB$ respectively. Points $S$, $A_1$, $B_1$, $C_1$, $D_1$ are distinct and lie on a sphere. Prove that points $A_1$, $B_1$, $C_1$ and $D_1$ are coplanar.

2015 Sharygin Geometry Olympiad Finals 10.8 

Does there exist a rectangle which can be divided into a regular hexagon with sidelength $1$ and several congruent right-angled triangles with legs $1$ and $\sqrt{3}$?

2015-2016 First Round

2016 Sharygin Geometry Olympiad First Round p1 grade 8

A trapezoid $ABCD$ with bases $AD$ and $BC$ is such that $AB = BD$. Let $M$ be the midpoint of $DC$. Prove that $\angle MBC$ = $\angle BCA$.

2016 Sharygin Geometry Olympiad First Round p2 grade 8
Mark three nodes on a cellular paper so that the semiperimeter of the obtained triangle would be equal to the sum of its two smallest medians.

2016 Sharygin Geometry Olympiad First Round p3 grade 8
Let $AH_1, BH_2$ be two altitudes of an acute-angled triangle $ABC, D$ be the projection of $H_1$ to $AC, E$ be the projection of $D$ to $AB, F$ be the common point of $ED$ and $AH_1$. Prove that $H_2F // BC$.

2016 Sharygin Geometry Olympiad First Round p4 grade 8
In quadrilateral $ABCD$ , $\angle B = \angle D = 90^o$ and $AC = BC + DC$. Point $P$ of ray $BD$ is such that $BP = AD$. Prove that line $CP$ is parallel to the bisector of angle $ABD$.

2016 Sharygin Geometry Olympiad First Round p5 grade 8
In quadrilateral $ABCD$, $AB = CD, M$ and $K$ are the midpoints of $BC$ and $AD$. Prove that the angle between $MK$ and $AC$ is equal to the half-sum of angles $BAC$ and $DCA$.

2016 Sharygin Geometry Olympiad First Round p6 grade 8
Let $M$ be the midpoint of side $AC$ of triangle $ABC,MD$ and $ME$ be the perpendiculars
from $M$ to $AB$ and $BC$ respectively. Prove that the distance between the circumcenters
of triangles $ABE$ and $BCD$ is equal to $AC /4$.

2016 Sharygin Geometry Olympiad First Round p7 grades 8-9

Let all distances between the vertices of a convex $n$-gon ($n > 3$) be different.
a) A vertex is called uninteresting if the closest vertex is adjacent to it. What is the minimal possible number of uninteresting vertices (for a given $n$)?
b) A vertex is called unusual if the farthest vertex is adjacent to it. What is the maximal possible number of unusual vertices (for a given $n$)?

2016 Sharygin Geometry Olympiad First Round p8 grades 8-9
Let $ABCDE$ be an inscribed pentagon such that  $\angle B +  \angle E =  \angle C +  \angle D$. Prove that $\angle CAD < \pi / 3 <  \angle A$.

2016 Sharygin Geometry Olympiad First Round p9 grades 8-9
Let $ABC$ be a right-angled triangle and $CH$ be the altitude from its right angle $C$. Points $O_1$ and $O_2$ are the incenters of triangles $ACH$ and $BCH$ respectively,  $P_1$ and $P_2$ are the touching points of their incircles with $AC$ and $BC$. Prove that lines $O_1P_1$ and $O_2P_2$ meet on $AB$.

2016 Sharygin Geometry Olympiad First Round p10 grades 8-9
Point $X$ moves along side $AB$ of triangle $ABC$, and point $Y$ moves along its circumcircle in such a way that line $XY$ passes through the midpoint of arc $AB$. Find the locus of the circumcenters of triangles $IXY$ , where I is the incenter of $ABC$.

2016 Sharygin Geometry Olympiad First Round p11 grades 8-10
Restore a triangle $ABC$ by vertex $B$, the centroid and the common point of the symmedian from $B$ with the circumcircle.

2016 Sharygin Geometry Olympiad First Round p12 grades 9-10
Let $BB_1$ be the symmedian of a nonisosceles acute-angled triangle $ABC$. Ray $BB_1$ meets the circumcircle of $ABC$ for the second time at point $L$. Let $AH_A, BH_B, CH_C$ be the altitudes of triangle $ABC$. Ray $BH_B$ meets the circumcircle of $ABC$ for the second time at point $T$. Prove that $H_A, H_C, T, L$ are concyclic.

2016 Sharygin Geometry Olympiad First Round p13 grades 9-10
Given are a triangle $ABC$ and a line $\ell$ meeting $BC, AC, AB$ at points $L_a, L_b, L_c$ respectively. The perpendicular from $L_a$ to $BC$ meets AB and AC at points $A_B$ and $A_C$ respectively. Point $O_a$ is the circumcenter of triangle $AA_bA_c$. Points $O_b$ and $O_c$ are defined similarly. Prove that $O_a, O_b$ and $O_c$ are collinear.

2016 Sharygin Geometry Olympiad First Round p14 grades 9-11
Let a triangle $ABC$ be given. Consider the circle touching its circumcircle at $A$ and touching externally its incircle at some point $A_1$. Points $B_1$ and $C_1$ are defined similarly.
a) Prove that lines $AA_1, BB_1$ and $CC1$ concur.
b) Let $A_2$ be the touching point of the incircle with $BC$. Prove that lines $AA_1$ and $AA_2$ are symmetric about the bisector of angle $\angle A$.

2016 Sharygin Geometry Olympiad First Round p15 grades 9-11
Let $O, M, N$ be the circumcenter, the centroid and the Nagel point of a triangle. Prove that angle $MON$ is right if and only if one of the triangle’s angles is equal to $60^o$.

2016 Sharygin Geometry Olympiad First Round p16 grades 9-11
Let $BB_1$ and $CC_1$ be altitudes of triangle $ABC$. The tangents to the circumcircle of $AB_1C_1$ at $B_1$ and $C_1$ meet AB and $AC$ at points $M$ and $N$ respectively. Prove that the common point of circles $AMN$ and $AB_1C_1$ distinct from $A$ lies on the Euler line of $ABC$.

2016 Sharygin Geometry Olympiad First Round p17 grades 9-11
Let $D$ be an arbitrary point on side $BC$ of triangle $ABC$. Circles $\omega_1$ and $\omega_2$ pass through $A$ and $D$ in such a way that $BA$ touches $\omega_1$ and $CA$ touches $\omega_2$. Let $BX$ be the second tangent from $B$ to $\omega_1$, and $CY$ be the second tangent from $C$ to $\omega_2$. Prove that the circumcircle of triangle $XDY$ touches $BC$.

2016 Sharygin Geometry Olympiad First Round p18 grades 9-11
Let $ABC$ be a triangle with $\angle C = 90^o$, and $K, L$ be the midpoints of the minor arcs $AC$ and $BC$ of its circumcircle. Segment $KL$ meets $AC$ at point $N$. Find angle $NIC$ where $I$ is the incenter of $ABC$.

2016 Sharygin Geometry Olympiad First Round p19 grades 9-11
Let $ABCDEF$ be a regular hexagon. Points $P$ and $Q$ on tangents to its circumcircle at $A$ and $D$ respectively are such that $PQ$ touches the minor arc $EF$ of this circle. Find the angle between $PB$ and $QC$.

2016 Sharygin Geometry Olympiad First Round p20 grades 10-11

The incircle $\omega$ of a triangle $ABC$ touches $BC, AC$ and $AB$ at points $A_0, B_0$ and $C_0$ respectively. The bisectors of angles $B$ and $C$ meet the perpendicular bisector to segment $AA_0$ at points $Q$ and $P$ respectively. Prove that $PC_0$ and $QB_0$ meet on $\omega$ .

2016 Sharygin Geometry Olympiad First Round p21 grades 10-11
The areas of rectangles $P$ and $Q$ are equal, but the diagonal of $P$ is greater. Rectangle $Q$ can be covered by two copies of $P$. Prove that $P$ can be covered by two copies of $Q$.

2016 Sharygin Geometry Olympiad First Round p22 grades 10-11
Let $M_A, M_B, M_C$ be the midpoints of the sides of a nonisosceles triangle $ABC$.
Points $H_A,H_B,H_C$ lying on the correspondent sides and distinct from $M_A, M_B, M_C$ are
such that $M_AH_B = M_AHC, M_BH_A = M_BH_C, M_CH_A = M_CH_B$. Prove that $H_A, H_B, H_C$ are the bases of the altitudes of $ABC$.

2016 Sharygin Geometry Olympiad First Round p23 grades 10-11
A sphere touches all edges of a tetrahedron. Let $a, b, c$ and d be the segments of the tangents to the sphere from the vertices of the tetrahedron. Is it true that that some of these segments necessarily form a triangle?

(It is not obligatory to use all segments. The side of the triangle can be formed by two segments)

2016 Sharygin Geometry Olympiad First Round p24 grade 11

A sphere is inscribed into a prism $ABCA'B'C'$ and touches its lateral faces $BCC'B', CAA'C', ABB'A'$ at points $A_o, B_o, C_o$ respectively. It is known that $\angle A_oBB' = \angle B_oCC' =\angle C_oAA'$.

2015-2016 Final Round

grade 8

2016 Sharygin Geometry Olympiad Finals 8.1
An altitude $AH$ of triangle $ABC$ bisects a median $BM$. Prove that the medians of triangle $ABM$ are sidelengths of a right-angled triangle.

by Yu.Blinkov

2016 Sharygin Geometry Olympiad Finals 8.p2
A circumcircle of triangle $ABC$ meets the sides $AD$ and $CD$ of a parallelogram $ABCD$ at points $K$ and $L$ respectively. Let $M$ be the midpoint of arc $KL$ not containing $B$. Prove that $DM \perp AC$.

by E.Bakaev

2016 Sharygin Geometry Olympiad Finals 8.3 
A trapezoid $ABCD$ and a line $\ell$ perpendicular to its bases $AD$ and $BC$ are given. A point $X$ moves along  $\ell$. The perpendiculars from $A$ to $BX$ and from $D$ to $CX$ meet at point $Y$ . Find the locus of $Y$ .

by D.Prokopenko

2016 Sharygin Geometry Olympiad Finals 8.4
Is it possible to dissect a regular decagon along some of its diagonals so that the resulting parts can form two regular polygons?

by N.Beluhov

2016 Sharygin Geometry Olympiad Finals 8.5
Three points are marked on the transparent sheet of paper. Prove that the sheet can be folded along some line in such a way that these points form an equilateral triangle.

by A.Khachaturyan

2016 Sharygin Geometry Olympiad Finals 8.6
A triangle ABC with $\angle A = 60^o$ is given. Points $M$ and $N$ on $AB$ and $AC$ respectively are such that the circumcenter of $ABC$ bisects segment $MN$. Find the ratio $AN:MB$.

by E.Bakaev

2016 Sharygin Geometry Olympiad Finals 8.7
Diagonals of a quadrilateral ABCD are equal and meet at point O. The perpendicular bisectors to segments AB and CD meet at point P, and the perpendicular bisectors to BC and AD meet at point Q. Find angle POQ.

by A.Zaslavsky

2016 Sharygin Geometry Olympiad Finals 8.8 

A criminal is at point $X$, and three policemen at points $A, B$ and $C$ block him up, i.e. the point $X$ lies inside the triangle $ABC$. Each evening one of the policemen is replaced in the following way: a new policeman  takes the position equidistant from three former policemen, after this one of the former policemen goes away so that three remaining policemen block up the criminal too. May the policemen after some time occupy again the points $A, B$ and $C$ (it is known that at any moment $X$ does not lie on a side of the triangle)?

by V.Protasov

grade 9

2016 Sharygin Geometry Olympiad Finals 9.1

The diagonals of a parallelogram $ABCD$ meet at point $O$. The tangent to the circumcircle of triangle $BOC$ at $O$ meets ray $CB$ at point $F$. The circumcircle of triangle $FOD$ meets $BC$ for the second time at point $G$. Prove that $AG=AB$.

2016 Sharygin Geometry Olympiad Finals 9.2
Let $H$ be the orthocenter of an acute-angled triangle $ABC$. Point $X_A$ lying on the tangent at $H$ to the circumcircle of triangle $BHC$ is such that $AH=AX_A$ and $X_A \not= H$. Points $X_B,X_C$ are defined similarly. Prove that the triangle $X_AX_BX_C$ and the orthotriangle of $ABC$ are similar.

2016 Sharygin Geometry Olympiad Finals 9.3 
Let $O$ and $I$ be the circumcenter and incenter of triangle $ABC$. The perpendicular from $I$ to $OI$ meets $AB$ and the external bisector of angle $C$ at points $X$ and $Y$ respectively. In what ratio does $I$ divide the segment $XY$?

2016 Sharygin Geometry Olympiad Finals 9.4
One hundred and one beetles are crawling in the plane. Some of the beetles are friends. Every one hundred beetles can position themselves so that two of them are friends if and only if they are at unit distance from each other. Is it always true that all one hundred and one beetles can do the same?

2016 Sharygin Geometry Olympiad Finals 9.5
The center of a circle $\omega_2$ lies on a circle $\omega_1$. Tangents $XP$ and $XQ$ to $\omega_2$ from an arbitrary point $X$ of $\omega_1$ ($P$ and $Q$ are the touching points) meet $\omega_1$ for the second time at points $R$ and $S$. Prove that the line $PQ$ bisects the segment $RS$.

2016 Sharygin Geometry Olympiad Finals 9.6 
The sidelines $AB$ and $CD$ of a trapezoid meet at point $P$, and the diagonals of this trapezoid meet at point $Q$. Point $M$ on the smallest base $BC$ is such that $AM=MD$. Prove that $\angle PMB=\angle QMB$.

2016 Sharygin Geometry Olympiad Finals 9.7
The sidelines $AB$ and $CD$ of a trapezoid meet at point $P$, and the diagonals of this trapezoid meet at point $Q$. Point $M$ on the smallest base $BC$ is such that $AM=MD$. Prove that $\angle PMB=\angle QMB$.

2016 Sharygin Geometry Olympiad Finals 9.8

The diagonals of a cyclic quadrilateral meet at point $M$. A circle $\omega$ touches segments $MA$ and $MD$ at points $P,Q$ respectively and touches the circumcircle of $ABCD$ at point $X$. Prove that $X$ lies on the radical axis of circles $ACQ$ and $BDP$.

by Ivan Frolov

grade 10

2016 Sharygin Geometry Olympiad Finals 10.1

A line parallel to the side $BC$ of a triangle $ABC$ meets the sides $AB$ and $AC$ at points $P$ and $Q$, respectively. A point $M$ is chosen inside the triangle $APQ$. The segments $MB$ and $MC$ meet the segment $PQ$ at points $E$ and $F$, respectively. Let $N$ be the second intersection point of the circumcircles of the triangles $PMF$ and $QME$. Prove that the points $A,M,N$ are collinear.

2016 Sharygin Geometry Olympiad Finals 10.2
Let $I$ and $I_a$ be the incenter and excenter (opposite vertex $A$) of a triangle $ABC$, respectively. Let $A'$ be the point on its circumcircle opposite to $A$, and $A_1$ be the foot of the altitude from $A$. Prove that $\angle IA_1I_a=\angle IA'I_a$.

 by Pavel Kozhevnikov 

2016 Sharygin Geometry Olympiad Finals 10.3 
Assume that the two triangles $ABC$ and $A'B'C'$ have the common incircle and the common circumcircle. Let a point $P$ lie inside both the triangles. Prove that the sum of the distances from $P$ to the sidelines of triangle $ABC$ is equal to the sum of distances from $P$ to the sidelines of triangle $A'B'C'$.

2016 Sharygin Geometry Olympiad Finals 10.4
The Devil and the Man play a game. Initially, the Man pays some cash $s$ to the Devil. Then he lists some $97$ triples $\{i,j,k\}$ consisting of positive integers not exceeding $100$. After that, the Devil draws some convex polygon $A_1A_2...A_{100}$ with area $100$ and pays to the Man, the sum of areas of all triangles $A_iA_jA_k$. Determine the maximal value of $s$ which guarantees that the Man receives at least as much cash as he paid.

by Nikolai Beluhov, Bulgaria 

2016 Sharygin Geometry Olympiad Finals 10.5
Does there exist a convex polyhedron having equal number of edges and diagonals?

(A diagonal of a polyhedron is a segment through two vertices not lying on the same face)

2016 Sharygin Geometry Olympiad Finals 10.6 
A triangle $ABC$ is given. The point $K$ is the base of the external bisector of angle $A$. The point $M$ is the midpoint of the arc $AC$ of the circumcircle. The point $N$ on the bisector of angle $C$ is such that $AN \parallel BM$. Prove that the points $M,N,K$ are collinear.

by Ilya Bogdanov

2016 Sharygin Geometry Olympiad Finals 10.7 
Restore a triangle by one of its vertices, the circumcenter and the Lemoine's point.

(The Lemoine's point is the intersection point of the reflections of the medians in the correspondent angle bisectors)

2016 Sharygin Geometry Olympiad Finals 10.8
Let $ABC$ be a non-isosceles triangle, let $AA_1$ be its angle bisector and $A_2$ be the touching point of the incircle with side $BC$. The points $B_1,B_2,C_1,C_2$ are defined similarly. Let $O$ and $I$ be the circumcenter and the incenter of triangle $ABC$. Prove that the radical center of the circumcircle of the triangles $AA_1A_2, BB_1B_2, CC_1C_2$ lies on the line $OI$.

2016-2017 First Round

2017 Sharygin Geometry Olympiad First Round p1 grade 8

Mark on a cellular paper four nodes forming a convex quadrilateral with the sidelengths equal to four different primes.

                                                                                                                                           

 by A.Zaslavsky

2017 Sharygin Geometry Olympiad First Round p2 grade 8

A circle cuts off four right-angled triangles from rectangle $ABCD$.Let $A_0, B_0, C_0$ and $D_0$ be the midpoints of the correspondent hypotenuses. Prove that $A_0C_0 = B_0D_0$

by L.Shteingarts

2017 Sharygin Geometry Olympiad First Round p3 grade 8

Let $I$ be the incenter of triangle $ABC$; $H_B, H_C$ the orthocenters of triangles $ACI$ and $ABI$ respectively; $K$ the touching point of the incircle with the side $BC$. Prove that $H_B, H_C$ and K are collinear.

by M.Plotnikov

2017 Sharygin Geometry Olympiad First Round p4 grade 8

A triangle $ABC$ is given. Let $C'$ be the vertex of an isosceles triangle $ABC'$ with $\angle C' = 120^{\circ}$ constructed on the other side of $AB$ than $C$, and $B'$ be the vertex of an equilateral triangle $ACB'$ constructed on the same side of $AC$ as $ABC$. Let $K$ be the midpoint of $BB'$. Find the angles of triangle $KCC'$.

by A.Zaslavsky

2017 Sharygin Geometry Olympiad First Round p5 grades 8-9

A segment $AB$ is fixed on the plane. Consider all acute-angled triangles with side $AB$. Find the locus of
а) the vertices of their greatest angles,
b) their incenters.

2017 Sharygin Geometry Olympiad First Round p6 grades 8-9
Let $ABCD$ be a convex quadrilateral with $AC = BD = AD$; $E$ and $F$ the midpoints of $AB$ and $CD$ respectively; $O$ the common point of the diagonals.Prove that $EF$ passes through the touching points of the incircle of triangle $AOD$ with $AO$ and $OD$

by N.Moskvitin

2017 Sharygin Geometry Olympiad First Round p7 grades 8-9
The circumcenter of a triangle lies on its incircle. Prove that the ratio of its greatest and smallest sides is less than two.

by B.Frenkin

2017 Sharygin Geometry Olympiad First Round p8 grades 8-9
Let $AD$ be the base of trapezoid $ABCD$. It is known that the circumcenter of triangle $ABC$ lies on $BD$. Prove that the circumcenter of triangle $ABD$ lies on $AC$.

by Ye.Bakayev

2017 Sharygin Geometry Olympiad First Round p9 grades 8-9

Let $C_0$ be the midpoint of hypotenuse $AB$ of triangle $ABC$, $AA_1, BB_1$ the bisectors of this triangle; $I$ its incenter. Prove that the lines $C_0I$ and $A_1B_1$ meet on the altitude from $C$.

by A.Zaslavsky 

2017 Sharygin Geometry Olympiad First Round p10 grades 8-10

Points $K$ and $L$ on the sides $AB$ and $BC$ of parallelogram $ABCD$ are such that $\angle AKD = \angle CLD$. Prove that the circumcenter of triangle $BKL$ is equidistant from $A$ and $C$.

 by I.I.Bogdanov 

2017 Sharygin Geometry Olympiad First Round p11 grades 8-11

A finite number of points is marked on the plane. Each three of them are not collinear. A circle is circumscribed around each triangle with marked vertices. Is it possible that all centers of these circles are also marked?

 by A.Tolesnikov

2017 Sharygin Geometry Olympiad First Round p12 grades 9-10

Let $AA_1 , CC_1$ be the altitudes of triangle $ABC, B_0$ the common point of the altitude from $B$ and the circumcircle of $ABC$; and $Q$ the common point of the circumcircles of $ABC$ and $A_1C_1B_0$, distinct from $B_0$. Prove that $BQ$ is the symmedian of $ABC$.

by D.Shvetsov

2017 Sharygin Geometry Olympiad First Round p13 grades 9-11

Two circles pass through points $A$ and $B$. A third circle touches both these circles and meets $AB$ at points $C$ and $D$. Prove that the tangents to this circle at these points are parallel to the common tangents of two given circles.

by A.Zaslavsky

2017 Sharygin Geometry Olympiad First Round p14 grades 9-11

Let points $B$ and $C$ lie on the circle with diameter $AD$ and center $O$ on the same side of $AD$. The circumcircles of triangles $ABO$ and $CDO$ meet $BC$ at points $F$ and $E$ respectively. Prove that $R^2 = AF.DE$, where $R$ is the radius of the given circle.

by N.Moskvitin 

2017 Sharygin Geometry Olympiad First Round p15 grades 9-11

Let $ABC$ be an acute-angled triangle with incircle $\omega$ and incenter $I$. Let $\omega$ touch $AB, BC$ and $CA$ at points $D, E, F$ respectively. The circles $\omega_1$ and $\omega_2$ centered at $J_1$ and $J_2$ respectively are inscribed into A$DIF$ and $BDIE$. Let $J_1J_2$ intersect $AB$ at point $M$. Prove that $CD$ is perpendicular to $IM$.

2017 Sharygin Geometry Olympiad First Round p16 grades 9-11

The tangents to the circumcircle of triangle $ABC$ at $A$ and $B$ meet at point $D$.  The circle passing through the projections of $D$ to $BC, CA, AB$, meet $AB$ for the second time at point $C'$. Points $A', B'$ are defined similarly. Prove that $AA', BB', CC'$ concur.

2017 Sharygin Geometry Olympiad First Round p17 grades 9-11

Using a compass and a ruler, construct a point $K$ inside an acute-angled triangle $ABC$ so that $\angle KBA = 2\angle KAB$ and $\angle KBC = 2\angle KCB$.

2017 Sharygin Geometry Olympiad First Round p18 grades 9-11

Let $L$ be the common point of the symmedians of triangle $ABC$, and $BH$ be its altitude. It is known that $\angle ALH = 180^o -2\angle A$. Prove that $\angle CLH = 180^o - 2\angle C$.

2017 Sharygin Geometry Olympiad First Round p19 grades 10-11

 Let cevians $AA', BB'$ and $CC'$ of triangle $ABC$ concur at point $P.$ The circumcircle of triangle $PA'B'$ meets $AC$ and $BC$ at points $M$ and $N$ respectively, and the circumcircles of triangles $PC'B'$ and $PA'C'$ meet $AC$ and $BC$ for the second time respectively at points $K$ and $L$. The line $c$ passes through the midpoints of segments $MN$ and $KL$. The lines $a$ and $b$ are defined similarly. Prove that $a$, $b$ and $c$ concur.

2017bSharygin Geometry Olympiad First Round p20 grades 10-11

Given a right-angled triangle $ABC$ and two perpendicular lines $x$ and $y$ passing through the vertex $A$ of its right angle. For an arbitrary point $X$ on $x$ define $y_B$ and $y_C$ as the reflections of $y$ about $XB$ and $XC$ respectively. Let $Y$ be the common point of $y_b$ and $y_c$. Find the locus of $Y$ (when $y_b$ and $y_c$ do not coincide).

2017 Sharygin Geometry Olympiad First Round p 21 grades 10-11

A convex hexagon is circumscribed about a circle of radius $1$. Consider the three segments joining the midpoints of its opposite sides. Find the greatest real number $r$ such that the length of at least one segment is at least $r.$

2017 Sharygin Geometry Olympiad First Round p22 grades 10-11

Let $P$ be an arbitrary point on the diagonal $AC$ of cyclic quadrilateral $ABCD$, and $PK, PL, PM, PN, PO$ be the perpendiculars from $P$ to $AB, BC, CD, DA, BD$ respectively. Prove that the distance from $P$ to $KN$ is equal to the distance from $O$ to $ML$.

2017 Sharygin Geometry Olympiad First Round p23 grades 10-11

Let a line $m$ touch the incircle of triangle $ABC$. The lines passing through the incenter $I$ and perpendicular to $AI, BI, CI$ meet $m$ at points $A', B', C'$ respectively. Prove that $AA', BB'$ and $CC'$ concur.

2017 Sharygin Geometry Olympiad First Round p24 grade 11

Two tetrahedrons are given. Each two faces of the same tetrahedron are not similar, but each face of the first tetrahedron is similar to some face of the second one. Does this yield that these tetrahedrons are similar?

2016-2017 Final Round

grade 8

2017 Sharygin Geometry Olympiad Finals 8.1 

Let $ABCD$ be a cyclic quadrilateral with $AB=BC$ and $AD = CD$. A point $M$ lies on the minor arc $CD$ of its circumcircle. The lines $BM$ and $CD$ meet at point $P$, the lines $AM$  and $BD$ meet at point $Q$. Prove that $PQ \parallel AC$.

2017 Sharygin Geometry Olympiad Finals 8.2

Let $H$ and $O$ be the orthocenter and circumcenter of an acute-angled triangle $ABC$, respectively. The perpendicular bisector of $BH$ meets $AB$ and $BC$ at points $A_1$ and $C_1$, respectively. Prove that $OB$ bisects the angle $A_1OC_1$.

2017 Sharygin Geometry Olympiad Finals 8.3

Let $AD, BE$ and $CF$ be the medians of triangle $ABC$. The points $X$ and $Y$ are the reflections of $F$ about $AD$ and $BE$, respectively. Prove that the circumcircles of triangles $BEX$ and $ADY$ are concentric.

2017 Sharygin Geometry Olympiad Finals 8.4 

Alex dissects a paper triangle into two triangles. Each minute after this he dissects one of obtained triangles into two triangles. After some time (at least one hour) it appeared that all obtained triangles were congruent. Find all initial triangles for which this is possible.

2017 Sharygin Geometry Olympiad Finals 8.5

A square $ABCD$ is given. Two circles are inscribed into angles $A$ and $B$, and the sum of their diameters is equal to the sidelength of the square. Prove that one of their common tangents passes through the midpoint of $AB$.

2017 Sharygin Geometry Olympiad Finals 8.6

A median of an acute-angled triangle dissects it into two triangles. Prove that each of them can be covered by a semidisc congruent to a half of the circumdisc of the initial triangle.

2017 Sharygin Geometry Olympiad Finals 8.7

Let $A_1A_2 \dots A_{13}$ and $B_1B_2 \dots B_{13}$ be two regular $13$-gons in the plane such that the points $B_1$ and $A_{13}$ coincide and lie on the segment $A_1B_{13}$, and both polygons lie in the same semiplane with respect to this segment. Prove that the lines $A_1A_9, B_{13}B_8$ and $A_8B_9$ are concurrent.

2017 Sharygin Geometry Olympiad Finals 8.8 

Let $ABCD$ be a square, and let $P$ be a point on the minor arc $CD$ of its circumcircle. The lines $PA, PB$ meet the diagonals $BD, AC$ at points $K, L$ respectively. The points $M, N$ are the projections of $K, L$ respectively to $CD$, and $Q$ is the common point of lines $KN$ and $ML$. Prove that $PQ$ bisects the segment $AB$.

grade 9

2017 Sharygin Geometry Olympiad Finals 9.1

Let $ABC$ be a regular triangle. The line passing through the midpoint of $AB$ and parallel to $AC$ meets the minor arc $AB$ of the circumcircle at point $K$. Prove that the ratio $AK:BK$ is equal to the ratio of the side and the diagonal of a regular pentagon.

2017 Sharygin Geometry Olympiad Finals 9.2

Let $I$ be the incenter of a triangle $ABC$, $M$ be the midpoint of $AC$, and $W$ be the midpoint of arc $AB$ of the circumcircle not containing $C$. It is known that $\angle AIM = 90^\circ$. Find the ratio $CI:IW$.

2017 Sharygin Geometry Olympiad Finals 9.3

The angles $B$ and $C$ of an acute-angled​ triangle $ABC$ are greater than $60^\circ$. Points $P,Q$ are chosen on the sides $AB,AC$ respectively so that the points $A,P,Q$ are concyclic with the orthocenter $H$ of the triangle $ABC$. Point $K$ is the midpoint of $PQ$. Prove that $\angle BKC > 90^\circ$.

 by A. Mudgal

2017 Sharygin Geometry Olympiad Finals 9.4

Points $M$ and $K$ are chosen on lateral sides $AB,AC$ of an isosceles triangle $ABC$ and point $D$ is chosen on $BC$ such that $AMDK$ is a parallelogram. Let the lines $MK$ and $BC$ meet at point $L$, and let $X,Y$ be the intersection points of $AB,AC$ with the perpendicular line from $D$ to $BC$. Prove that the circle with center $L$ and radius $LD$ and the circumcircle of triangle $AXY$ are tangent.

2017 Sharygin Geometry Olympiad Finals 9.5 

Let $BH_b, CH_c$ be altitudes of an acute-angled triangle $ABC$. The line $H_bH_c$ meets the circumcircle of $ABC$ at points $X$ and $Y$. Points $P,Q$ are the reflections of $X,Y$ about $AB,AC$ respectively. Prove that  $PQ \parallel BC$.

by Pavel Kozhevnikov

2017 Sharygin Geometry Olympiad Finals 9.6 

Let $ABC$ be a right-angled triangle ($\angle C = 90^\circ$) and $D$ be the midpoint of an altitude from C. The reflections of the line $AB$ about $AD$ and $BD$, respectively, meet at point $F$. Find the ratio $S_{ABF}:S_{ABC}$.

Note: $S_{\alpha}$ means the area of $\alpha$.

2017 Sharygin Geometry Olympiad Finals 9.7

Let $a$ and $b$ be parallel lines with $50$ distinct points marked on $a$ and $50$ distinct points marked on $b$. Find the greatest possible number of acute-angled triangles all of whose vertices are marked.

2017 Sharygin Geometry Olympiad Finals 9.8 

Let $AK$ and $BL$ be the altitudes of an acute-angled triangle $ABC$, and let $\omega$ be the excircle of $ABC$ touching side $AB$. The common internal tangents to circles $CKL$ and $\omega$ meet $AB$ at points $P$ and $Q$. Prove that $AP =BQ$.

by I.Frolov

grade 10

2017 Sharygin Geometry Olympiad Finals 10.1

Let $A$ and $B$ be the common points of two circles, and $CD$ be their common tangent ($C$ and
$D$ are the tangency points). Let $Oa, Ob$ be the circumcenters of triangles $CAD, CBD$ respectively. Prove that the midpoint of segment $OaOb$ lies on the line $AB$.

2017 Sharygin Geometry Olympiad Finals 10.p2

Prove that the distance from any vertex of an acute-angled triangle to the corresponding excenter is less than the sum of two greatest sidelengths.

2017 Sharygin Geometry Olympiad Finals 10.3

Let $ABCD$ be a convex quadrilateral, and let $\omega_A, \omega_B, \omega_C, \omega_D$ be the circumcircles of triangles $BCD, ACD, ABD, ABC$, respectively. Denote by $X_A$ the product of the power of $A$ with respect to $\omega_A$ and the area of triangle $BCD$. Define $X_B,X_C,X_D$ similarly. Prove that $X_A + X_B + X_C + X_D = 0$.

2017 Sharygin Geometry Olympiad Finals 10.4

A scalene triangle $ABC$ and its incircle $\omega$ are given. Using only a ruler and drawing at most eight lines, rays or segments, construct points $A', B', C'$ on $\omega$  such that the rays $B'C', C'A', A'B'$ pass through $A, B, C$, respectively.

2017 Sharygin Geometry Olympiad Finals 10.5

Let $BB', CC'$ be the altitudes of an acuteangled triangle $ABC$. Two circles passing through $A$
and $C'$ are tangent to $BC$ at points $P$ and $Q$. Prove that $A, B', P, Q$ are concyclic.

2017 Sharygin Geometry Olympiad Finals 10.6 

Let the insphere of a pyramid $SABC$ touch the faces $SAB, SBC, SCA$ at points $D, E, F$ respectively. Find all possible values of the sum of angles $SDA, SEB$ and $SFC$.

2017 Sharygin Geometry Olympiad Finals 10.7

A quadrilateral $ABCD$ is circumscribed around circle $\omega$ centered at $I$ and inscribed into
circle $\Gamma$ . The lines $AB$ and $CD$ meet at point $P$, the lines $BC$ and $AD$ meet at point $Q$. Prove that the circles $PIQ$ and $\Gamma$ are orthogonal.

2017 Sharygin Geometry Olympiad Finals 10.8
Suppose $S$ is a set of points in the plane, $|S|$ is even, no three points of $S$ are collinear. Prove that $S$ can be partitioned into two sets $S_1$ and $S_2$ so that their convex hulls have equal number of vertices.

2017-2018 First Round

2018 Sharygin Geometry Olympiad First Round p1 grade 8 

Three circles lie inside a square. Each of them touches externally two remaining circles. Also each circle touches two sides of the square. Prove that two of these circles are congruent.

2018 Sharygin Geometry Olympiad First Round p2 grade 8 

A cyclic quadrilateral $ABCD$ is given. The lines $AB$ and $DC$ meet at point $E$, and the lines $BC$ and $AD$ meet at point $F$. Let $I$  be the incenter of triangle $AED$, and a ray with origin $F$ be perpendicular to the bisector of angle $AID$. In what ratio does this ray dissect the angle $AFB$?

2018 Sharygin Geometry Olympiad First Round p3 grade 8

Let $AL$ be a bisector of triangle $ABC$, $D$ be its midpoint, and $E$ be the projection of $D$ to $AB$. It is known that $AC = 3AE$. Prove that $CEL$ is an isosceles triangle.

2018 Sharygin Geometry Olympiad First Round p4 grade 8

Let $ABCD$ be a cyclic quadrilateral. A point $P$ moves along the arc $AD$ which does not contain $B$ and $C$. A fixed line $\ell$, perpendicular to $BC$, meets the rays $BP, CP$ at points $B_0, C_0$ respectively. Prove that the tangent at $P$ to the circumcircle of triangle $PB_0C_0$ passes through some fixed point.

2018 Sharygin Geometry Olympiad First Round p5 grades 8-9 

The vertex $C$ of equilateral triangles $ABC$ and $CDE$ lies on the segment $AE$, and the vertices $B$ and $D$ lie on the same side with respect to this segment. The circumcircles of these triangles centered at $O_1$ and $O_2$ meet for the second time at point $F$. The lines $O_1O_2$ and $AD$ meet at point $K$. Prove that $AK = BF$.

2018 Sharygin Geometry Olympiad First Round p6 grades 8-9 

Let $CH$ be the altitude of a right-angled triangle $ABC$ ($\angle C = 90^o$) with $BC = 2AC$. Let $O_1, O_2$ and $O$ be the incenters of triangles $ACH, BCH$ and $ABC$ respectively, and $H_1, H_2, H_0$ be the projections of $O_1, O_2, O$ respectively to $AB$. Prove that $H_1H = HH_0 = H_0H_2$.

2018 Sharygin Geometry Olympiad First Round p7 grades 8-9 

Let $E$ be a common point of circles $w_1$ and $w_2$. Let $AB$ be a common tangent to these circles, and $CD$ be a line parallel to $AB$, such that $A$ and $C$ lie on $w_1, B$ and $D$ lie on $w_2$. The circles ABE and $CDE$ meet for the second time at point $F$. Prove that $F$ bisects one of arcs $CD$ of circle $CDE$.

2018 Sharygin Geometry Olympiad First Round p8 grades 8-9 

Restore a triangle $ABC$ by the Nagel point, the vertex $B$ and the foot of the altitude from this vertex.

2018 Sharygin Geometry Olympiad First Round p9 grades 8-9

A square is inscribed into an acute-angled triangle: two vertices of this square lie on the same side of the triangle and two remaining vertices lies on two remaining sides. Two similar squares are constructed for the remaining sides. Prove that three segments congruent to the sides of these squares can be the sides of an acute-angled triangle.

2018 Sharygin Geometry Olympiad First Round p10 grades 8-9

In the plane, $2018$ points are given such that all distances between them are different. For each point, mark the closest one of the remaining points. What is the minimal number of marked points?

2018 Sharygin Geometry Olympiad First Round p11 grades 8-9

Let $I$ be the incenter of a nonisosceles triangle $ABC$. Prove that there exists a unique pair of points $M, N$ lying on the sides $AC, BC$ respectively, such that $\angle AIM = \angle  BIN$ and $MN // AB$.

2018 Sharygin Geometry Olympiad First Round p12 grades 8-9

Let $BD$ be the external bisector of a triangle $ABC$ with $AB> BC, K$ and $K_1$ be the touching points of side $AC$ with the incicrle and the excircle centered at $I$ and $I_1$ respectively. The lines $BK$ and $DI_1$ meet at point $X$, and the lines $BK_1$ and $DI$ meet at point $Y$ . Prove that $XY \perp AC$.

2018 Sharygin Geometry Olympiad First Round p13 grades 9-11

Let $ABCD$ be a cyclic quadrilateral, and $M, N$ be the midpoints of arcs $AB$ and $CD$ respectively. Prove that $MN$ bisects the segment between the incenters of triangles $ABC$ and $ADC$.

2018 Sharygin Geometry Olympiad First Round p14 grades 9-11

Let $ABC$ be a right-angled triangle with $\angle C = 90^o, K, L, M$ be the midpoints of sides $AB, BC, CA$ respectively, and $N$ be a point of side $AB$. The line $CN$ meets $KM$ and $KL$ at points $P$ and $Q$ respectively. Points $S, T$ lying on $AC$ and $BC$ respectively are such that $APQS$ and $BPQT$ are cyclic quadrilaterals. Prove that

a) if $CN$ is a bisector, then $CN, ML$ and $ST$ concur;

b) if $CN$ is an altitude, then $ST$ bisects $ML$.

2018 Sharygin Geometry Olympiad First Round p15 grades 9-11

The altitudes $AH_1,BH_2,CH_3$ of an acute-angled triangle $ABC$ meet at point $H$. Points $P$ and $Q$ are the reflections of $H_2$ and $H_3$ with respect to $H$. The circumcircle of triangle $PH_1Q$ meets for the second time $BH_2$ and $CH_3$ at points $R$ and $S$. Prove that $RS$ is a medial line of triangle $ABC$.

2018 Sharygin Geometry Olympiad First Round p16 grades 9-11

Let $ABC$ be a triangle with $AB < BC$. The bisector of angle $C$ meets the line parallel to $AC$ and passing through $B$, at point $P$. The tangent at $B$ to the circumcircle of $ABC$ meets this bisector at point $R$. Let $R'$ be the reflection of R with respect to $AB$. Prove that $\angle R'PB =\angle RPA$.

2018 Sharygin Geometry Olympiad First Round p17 grades 10-11

Let each of circles $\alpha, \beta, \gamma$  touch two remaining circles externally, and all of them touch a circle $\Omega$  internally at points $A_1, B_1, C_1$ respectively. The common internal tangent to $\alpha$ and $\beta$ meets the arc $A_1B_1$ not containing $C_1$, at point $C_2$. Points $A_2, B_2$ are defined similarly. Prove that the lines $A1A2, B_1B_2, C_1C_2$ concur.

2018 Sharygin Geometry Olympiad First Round p18 grades 10-11

Let $C_1,A_1,B_1$ be points on sides $AB,BC,CA$ of triangle $ABC$, such that $AA_1,BB_1,CC_1$ concur. The rays $B_1A_1$ and $B_1C_1$ meet the circumcircle of the triangle atnpoints $A_2$ and $C_2$ respectively. Prove that $A,C,$ the common point of $A_2C_2$ and $BB_1$, and the midpoint of $A_2C_2$ are concyclic.

2018 Sharygin Geometry Olympiad First Round p19 grades 10-11

Let a triangle $ABC$ be given. On a ruler, three segments congruent to the sides of this triangle are marked. Using this ruler construct the orthocenter of the triangle formed by the tangency points of the sides of $ABC$ with its incircle.

2018 Sharygin Geometry Olympiad First Round p20 grades 10-11

Let the incircle of a nonisosceles triangle $ABC$ touch $AB$, $AC$ and$BC$ at points $D, E$ and $F$ respectively. The corresponding excircle touches the side $BC$ at point $N$. Let $T$ be the common point of $AN$ and the incircle, closest to $N$, and $K$ be the common point of $DE$ and $FT$. Prove that $AK//BC$.

2018 Sharygin Geometry Olympiad First Round p21 grades 10-11

In the plane, a line $\ell$ and a point $A$ outside it are given. Find the locus of the incenters of acute-angled triangles having the vertex $A$and the opposite side lying on  $\ell$ .

2018 Sharygin Geometry Olympiad First Round p22 grades 10-11

Six circles of unit radius lie in the plane so that the distance between the centers of any two of them is greater than d. What is the least value of d such that there always exists a straight line which does not intersect any of the circles and separates the circles into two groups of three?

2018 Sharygin Geometry Olympiad First Round p23 grades 10-11

The plane is divided into convex heptagons with diameters less than $1$. Prove that an arbitrary disc with radius $200$ intersects more than a billion of them.

2018 Sharygin Geometry Olympiad First Round p24 grades 10-11

A crystal of pyrite is a parallelepiped with dashed faces. The dashes on any two adjacent faces are perpendicular. Does there exist a convex polytope with the number of faces not equal to $6$, such that its faces can be dashed in such a manner?

2017 -2018 Final Round

grade 8

2018 Sharygin Geometry Olympiad Finals 8.1

The incircle of a right-angled triangle $ABC$ ($\angle C = 90^\circ$) touches $BC$ at point $K$. Prove that the chord of the incircle cut by line $AK$ is twice as large as the distance from $C$ to that line.

2018 Sharygin Geometry Olympiad Finals 8.2

A rectangle $ABCD$ and its circumcircle are given. Let $E$ be an arbitrary point  on the minor arc $BC$. The tangent to the circle at $B$ meets $CE$ at point $G$. The segments $AE$ and $BD$ meet at point $K$. Prove that $GK$ and $AD$ are perpendicular.

2018 Sharygin Geometry Olympiad Finals 8.3

Let $ABC$ be a triangle with $\angle A = 60^\circ$, and $AA', BB', CC'$ be its internal angle bisectors. Prove that $\angle B'A'C' \le 60^\circ$.

2018 Sharygin Geometry Olympiad Finals 8.4

Find all sets of six points in the plane, no three collinear, such that if we partition the set into two sets, then the obtained triangles are congruent.

2018 Sharygin Geometry Olympiad Finals 8.5

The side $AB$ of a square $ABCD$ is the base of an isosceles triangle $ABE$ such that $AE=BE$ lying outside the square. Let $M$ be the midpoint of $AE$, $O$ be the intersection of $AC$ and $BD$. $K$ is the intersection of $OM$ and $ED$. Prove that $EK=KO$.

2018 Sharygin Geometry Olympiad Finals 8.6

Suppose $ABCD$ and $A_1B_1C_1D_1$ be quadrilaterals with corresponding angles equal. Also $AB=A_1B_1$, $AC=A_1C_1$, $BD=B_1D_1$. Are the quadrilaterals necessarily congruent?

2018 Sharygin Geometry Olympiad Finals 8.7

Let $\omega_1,\omega_2$ be two circles centered at $O_1$ and $O_2$ and lying outside each other. Points $C_1$ and $C_2$ lie on these circles in the same semi plane with respect to $O_1O_2$. The ray $O_1C_1$ meets $\omega _2$ at $A_2,B_2$ and $O_2C_2$ meets $\omega_1$ at $A_1,B_1$. Prove that $\angle A_1O_1B_1=\angle A_2O_2B_2$ if and only if $C_1C_2||O_1O_2$.

2018 Sharygin Geometry Olympiad Finals 8.8

Let $I$ be the incenter of fixed triangle $ABC$, and $D$ be an arbitrary point on $BC$. The perpendicular bisector of $AD$ meets $BI,CI$ at $F$ and $E$ respectively. Find the locus of orthocenters of $\triangle IEF$ as $D$ varies.

grade 9

2018 Sharygin Geometry Olympiad Finals 9.1

Let $M$ be the midpoint of $AB$ in a right angled triangle $ABC$ with $\angle C = 90^\circ$. A circle passing through $C$ and $M$ meets segments $BC, AC$ at  $P, Q$ respectively. Let $c_1, c_2$ be the circles with centers $P, Q$ and radii $BP, AQ$ respectively. Prove that $c_1, c_2$ and the circumcircle of $ABC$ are concurrent.

2018 Sharygin Geometry Olympiad Finals 9.2

A triangle $ABC$ is given. A circle $\gamma$ centered at $A$ meets segments $AB$ and $AC$. The common chord of $\gamma$ and the circumcircle of $ABC$ meets $AB$ and $AC$ at $X$ and $Y$, respectively. The segments $CX$ and $BY$ meet $\gamma$ at point $S$ and $T$, respectively. The circumcircles of triangles $ACT$ and $BAS$ meet at points $A$ and $P$. Prove that $CX, BY$ and $AP$ concur.

2018 Sharygin Geometry Olympiad Finals 9.3

The vertices of a triangle $DEF$ lie on different sides of a triangle $ABC$. The lengths of the tangents from the incenter of $DEF$ to the excircles of $ABC$ are equal. Prove that $4S_{DEF} \ge S_{ABC}$.

Note: By $S_{XYZ}$ we denote the area of triangle $XYZ$.

2018 Sharygin Geometry Olympiad Finals 9.4

Let $BC$ be a fixed chord of a circle $\omega$. Let $A$ be a variable point on the major arc $BC$ of $\omega$. Let $H$ be the orthocenter of $ABC$. The points $D, E$ lie on $AB, AC$ such that $H$ is the midpoint of $DE$. $O_A$ is the circumcenter of $ADE$. Prove that as $A$ varies, $O_A$ lies on a fixed circle.

2018 Sharygin Geometry Olympiad Finals 9.5

Let $ABCD$ be a cyclic quadrilateral, $BL$ and $CN$ be the internal angle bisectors in triangles $ABD$ and $ACD$ respectively. The circumcircles of triangles $ABL$ and $CDN$ meet at points $P$ and $Q$. Prove that the line $PQ$ passes through the midpoint of the arc $AD$ not containing $B$.

2018 Sharygin Geometry Olympiad Finals 9.6

Let $ABCD$ be a circumscribed quadrilateral. Prove that the common point of the diagonals, the incenter of triangle $ABC$ and the centre of excircle of triangle $CDA$ touching the side $AC$ are collinear.

2018 Sharygin Geometry Olympiad Finals 9.7 

Let $B_1,C_1$ be the midpoints of sides $AC,AB$ of a triangle $ABC$ respectively. The tangents to the circumcircle at $B$ and $C$ meet the rays $CC_1,BB_1$ at points $K$ and $L$ respectively. Prove that $\angle BAK = \angle CAL$.

2018 Sharygin Geometry Olympiad Finals 9.8 

Consider a fixed regular $n$-gon of unit side. When a second regular $n$-gon of unit size rolls around the first one, one of its vertices successively pinpoints the vertices of a closed broken line $\kappa$ as in the figure. Let $A$ be the area of a regular $n$-gon of unit side, and let $B$ be the area of a regular $n$-gon of unit circumradius. Prove that the area enclosed by $\kappa$ equals $6A-2B$.

grade 10

2018 Sharygin Geometry Olympiad Finals 10.1

The altitudes $AH, CH$ of an acute-angled triangle $ABC$ meet the internal bisector of angle $B$ at points $L_1, P_1$, and the external bisector of this angle at points $L_2, P_2$. Prove that the orthocenters of triangles $HL_1P_1, HL_2P_2$ and the vertex $B$ are collinear.

2018 Sharygin Geometry Olympiad Finals 10.2

A fixed circle $\omega$ is inscribed into an angle with vertex $C$. An arbitrary circle passing through $C$, touches $\omega$ externally and meets the sides of the angle at points $A$ and $B$. Prove that the perimeters of all triangles $ABC$ are equal.

2018 Sharygin Geometry Olympiad Finals 10.3

A cyclic $n$-gon is given. The midpoints of all its sides are concyclic. The sides of the $n$-gon cut $n$ arcs of this circle lying outside the $n$-gon. Prove that these arcs can be coloured red and blue in such a way that the sum of the lengths of the red arcs is equal to the sum of the lengths of the blue arcs.

2018 Sharygin Geometry Olympiad Finals 10.4

We say that a finite set $S$ of red and green points in the plane is  separable if there exists a triangle $\delta$ such that all points of one colour lie strictly inside $\delta$ and all points of the other colour lie strictly outside of $\delta$. Let $A$ be a finite set of red and green points in the plane, in general position. Is it always true that if every $1000$ points in $A$ form a separable  set then $A$ is also separable?

2018 Sharygin Geometry Olympiad Finals 10.5

Let $\omega$ be the incircle of a triangle $ABC$. The line passing though the incenter $I$ and parallel to $BC$ meets $\omega$ at $A_b$ and $A_c$ ($A_b$ lies in the same semi plane with respect to $AI$ as $B$). The lines $BA_b$ and $CA_c$ meet at $A_1$. The points $B_1$ and $C_1$ are defined similarly. prove that $AA_1,BB_1,CC_1$ concur.

2018 Sharygin Geometry Olympiad Finals 10.6

Let $\omega$ be the circumcircle of $ABC$, and $KL$ be the diameter of $\omega$ passing through $M$ midpoint of $AB$ ($K,C$ lies on different sides of $AB$). A circle passing through $L$ and $M$ meets $CK$ at points $P$ and $Q$ ($Q$ lies on $KP$). Let $LQ$ meet the circumcircle of $KMQ$ again at $R$. Prove that $APBR$ is cyclic.

2018 Sharygin Geometry Olympiad Finals 10.7 

A convex quadrilateral $ABCD$ is circumscribed about a circle of radius $r$. What is the maximum value of $\frac{1}{AC^2}+\frac{1}{BD^2}$?

2018 Sharygin Geometry Olympiad Finals 10.8

Two triangles $ABC$ and $A'B'C'$ are given. The lines $AB$ and $A'B'$ meet at $C_1$ and the lines parallel to them and passing through $C$ and $C'$ meet at $C_2$. The points $A_1,A_2$, $B_1,B_2$ are defined similarly. Prove that $A_1A_2,B_1B_2,C_1C_1$ are either parallel or concurrent.

2018-2019 First Round

2019 Sharygin Geometry Olympiad First Round p1 grade 8
Let $AA_1$, $CC_1$ be the altitudes of $\ Delta ABC$, and $P$ be an arbitrary point of side $BC$. Point $Q$ on the line $AB$ is such that $QP = PC_1$, and point $R$ on the line $AC$ is such that $RP = CP$. Prove that $QA_1RA$ is a cyclic quadrilateral. 

2019 Sharygin Geometry Olympiad First Round p2 grade 8  

 The circle $\omega_1$ passes through the center $O$ of the circle $\omega_2$ and meets it at points $A$ and $B$. The circle $\omega_3$ centered at $A$ with radius $AB$ meets $\omega_1$ and $\omega_2$ at points $C$ and $D$ (distinct from $B$). Prove that $C, O, D$ are collinear. 

2019 Sharygin Geometry Olympiad First Round p3 grade 8 
The rectangle $ABCD$ lies inside a circle. The rays $BA$ and $DA$ meet this circle at points $A_1$ and $A_2$. Let $A_0$ be the midpoint of $A_1A_2$. Points $B_0$, $C_0, D_0$ are defined similarly. Prove that $A_0C_0 = B_0D_0$. 

2019 Sharygin Geometry Olympiad First Round p4 grade 8  

 The side $AB$ of $\ Delta ABC$ touches the corresponding excircle at point $T$. Let $J$ be the center of the excircle inscribed into $\angle A$, and $M$ be the midpoint of $AJ$. Prove that $MT = MC$.

2019 Sharygin Geometry Olympiad First Round p5 grades 8-9 
Let $A, B, C$ and $D$ be four points in general position, and $\omega$ be a circle passing through $B$ and $C$. A point $P$ moves along $\omega$. Let $Q$ be the common point of circles $\ odot (ABP)$ and $\ odot (PCD)$ distinct from $P$. Find the locus of points $Q$. 

2019 Sharygin Geometry Olympiad First Round p6 grades 8-9 

Two quadrilaterals $ABCD$ and $A_1B_1C_1D_1$ are mutually symmetric with respect to the point $P$. It is known that $A_1BCD$, $AB_1CD$ and $ABC_1D$ are cyclic quadrilaterals. Prove that the quadrilateral $ABCD_1$ is also cyclic

2019 Sharygin Geometry Olympiad First Round p7 grades 8-9 
Let $AH_A$, $BH_B$, $CH_C$ be the altitudes of the acute-angled $\ Delta ABC$. Let $X$ be an arbitrary point of segment $CH_C$, and $P$ be the common point of circles with diameters $H_CX$ and BC, distinct from $H_C$. The lines $CP$ and $AH_A$ meet at point $Q$, and the lines $XP$ and $AB$ meet at point $R$. Prove that $A, P, Q, R, H_B$ are concyclic.

2019 Sharygin Geometry Olympiad First Round p8 grades 8-9 

The circle $\omega_1$ passes through the vertex $A$ of the parallelogram $ABCD$ and touches the rays $CB, CD$. The circle $\omega_2$ touches the rays $AB, AD$ and touches $\omega_1$ externally at point $T$. Prove that $T$ lies on the diagonal $AC$ 

2019 Sharygin Geometry Olympiad First Round p9 grades 8-9 
Let $A_M$ be the midpoint of side $BC$ of an acute-angled $\ Delta ABC$, and $A_H$ be the foot of the altitude to this side. Points $B_M, B_H, C_M, C_H$ are defined similarly. Prove that one of the ratios $A_MA_H: A_HA, B_MB_H: B_HB, C_MC_H: ​​C_HC$ is equal to the sum of two remaining ratios 

2019 Sharygin Geometry Olympiad First Round p10 grades 8-9 

Let $N$ be the midpoint of arc $ABC$ of the circumcircle of $\ Delta ABC$, and $NP$, $NT$ be the tangents to the incircle of this triangle. The lines $BP$ and $BT$ meet the circumcircle for the second time at points $P_1$ and $T_1$ respectively. Prove that $PP_1 = TT_1$. 

2019 Sharygin Geometry Olympiad First Round p11 grades 8-9 
Morteza marks six points in the plane. He then calculates and writes down the area of ​​each triangle with vertices in these points ($20$ numbers). Is it possible that all of these numbers are integers, and that they add up to $2019$? 

2019 Sharygin Geometry Olympiad First Round p12 grades 8-11

 Let $A_1A_2A_3$ be an acute-angled triangle inscribed into a unit circle centered at $O$. The cevians from $A_i$ passing through $O$ meet the opposite sides at points $B_i$ $(i = 1, 2, 3)$ respectively. Find the minimum possible length of the longest of three segments $B_iO$. Find the maximum possible length of the shortest of three segments $B_iO$. 

2019 Sharygin Geometry Olympiad First Round p13 grades 9-10 
Let $ABC$ be an acute-angled triangle with altitude $AT = h$. The line passing through its circumcenter $O$ and incenter $I$ meets the sides $AB$ and $AC$ at points $F$ and $N$, respectively. It is known that $BFNC$ is a cyclic quadrilateral. Find the sum of the distances from the orthocenter of $ABC$ to its vertices. 

2019 Sharygin Geometry Olympiad First Round p14 grades 9-11 

 Let the side $AC$ of triangle $ABC$ touch the incircle and the corresponding excircle at points $K$ and $L$ respectively. Let $P$ be the projection of the incenter onto the perpendicular bisector of $AC$. It is known that the tangents to the circumcircle of triangle $BKL$ at $K$ and $L$ meet on the circumcircle of $ABC$. Prove that the lines $AB$ and $BC$ touch the circumcircle of triangle $PKL$.

2019 Sharygin Geometry Olympiad First Round p15 grades 9-11 
The incircle $\omega$ of triangle $ABC$ touches the sides $BC$, $CA$ and $AB$ at points $D$, $E$ and $F$ respectively . The perpendicular from $E$ to $DF$ meets $BC$ at point $X$, and the perpendicular from $F$ to $DE$ meets $BC$ at point $Y$. The segment $AD$ meets $\omega$ for the second time at point $Z$. Prove that the circumcircle of the triangle $XYZ$ touches $\omega$. 

2019 Sharygin Geometry Olympiad First Round p16 grades 9-11  

 Let $AH_1$ and $BH_2$ be the altitudes of triangle $ABC$. Let the tangent to the circumcircle of $ABC$ at $A$ meet $BC$ at point $S_1$, and the tangent at $B$ meet $AC$ at point $S_2$. Let $T_1$ and $T_2$ be the midpoints of $AS_1$ and $BS_2$ respectively. Prove that $T_1T_2$, $AB$ and $H_1H_2$ concur.

2019 Sharygin Geometry Olympiad First Round p17 grades 10-11 
Three circles $\omega_1$, $\omega_2$, $\omega_3$ are given. Let $A_0$ and $A_1$ be the common points of $\omega_1$ and $\omega_2$, $B_0$ and $B_1$ be the common points of $\omega_2$ and $\omega_3$, $C_0$ and $C_1$ be the common points of $\omega_3$ and $\omega_1$. Let $O_ {i, j, k}$ be the circumcenter of triangle $A_iB_jC_k$. Prove that the four lines of the form $O_ {ijk} O_ {1 - i, 1 - j, 1 - k}$ are concurrent or parallel.

2019 Sharygin Geometry Olympiad First Round p18 grades 10-11 
A quadrilateral $ABCD$ without parallel sidelines is circumscribed around a circle centered at $I$. Let $K, L, M$ and $N$ be the midpoints of $AB, BC, CD$ and $DA$ respectively. It is known that $AB \ cdot CD = 4IK \ cdot IM$. Prove that $BC \ cdot AD = 4IL \ cdot IN$. 

2019 Sharygin Geometry Olympiad First Round p19 grades 10-11 

Let $AL_a$, $BL_b$, $CL_c$ be the bisecors of triangle $ABC$. The tangents to the circumcircle of $ABC$ at $B$ and $C$ meet at point $K_a$, points $K_b$, $K_c$ are defined similarly. Prove that the lines $K_aL_a$, $K_bL_b$ and $K_cL_c$ concur.

2019 Sharygin Geometry Olympiad First Round p20 grades 10-11
Let $O$ be the circumcenter of triangle ABC, $H$ be its orthocenter, and $M$ be the midpoint of $AB$. The line $MH$ meets the line passing through $O$ and parallel to $AB$ at point $K$ lying on the circumcircle of $ABC$. Let $P$ be the projection of $K$ onto $AC$. Prove that $PH \ parallel BC$. 

2019 Sharygin Geometry Olympiad First Round p21 grades 10-11 

An ellipse $\ Gamma$ and its chord $AB$ are given. Find the locus of orthocenters of triangles $ABC$ inscribed into $\ Gamma$. 

2019 Sharygin Geometry Olympiad First Round p22 grades 10-11 
Let $AA_0$ be the altitude of the isosceles triangle $ABC ~ (AB = AC)$. A circle $\ gamma$ centered at the midpoint of $AA_0$ touches $AB$ and $AC$. Let $X$ be an arbitrary point of line $BC$. Prove that the tangents from $X$ to $\ gamma$ cut congruent segments on lines $AB$ and $AC$ 

2019 Sharygin Geometry Olympiad First Round p23 grades 10-11 
In the plane, let $a$, $b$ be two closed broken lines (possibly self-intersecting), and $K$, $L$, $M$, $N$ be four points. The vertices of $a$, $b$ and the points $K$ $L$, $M$, $N$ are in general position (ie no three of these points are collinear, and no three segments between them concur at an interior point). Each of segments $KL$ and $MN$ meets $a$ at an even number of points, and each of segments $LM$ and $NK$ meets $a$ at an odd number of points. Conversely, each of segments $KL$ and $MN$ meets $b$ at an odd number of points, and each of segments $LM$ and $NK$ meets $b$ at an even number of points. Prove that $a$ and $b$ intersect. 

2019 Sharygin Geometry Olympiad First Round p24 grade 11 
Two unit cubes have a common center. Is it always possible to number the vertices of each cube from $1$ to $8$ so that the distance between each pair of identically numbered vertices would be at most $4/5$? What about at most $13/16$?

2018 -2019 Final Round

grade 8

2019 Sharygin Geometry Olympiad Finals 8.1

A trapezoid with bases $AB$ and $CD$ is inscribed into a circle centered at $O$. Let $AP$ and $AQ$ be the tangents from $A$ to the circumcircle of triangle $CDO$. Prove that the circumcircle of triangle $APQ$ passes through the midpoint of $AB$.

2019 Sharygin Geometry Olympiad Finals 8.2 

A point $M$ inside triangle $ABC$ is such that $AM = AB / 2$ and $CM = BC / 2$. Points $C_0$ and $A_0$ lying on $AB$ and $CB$ respectively are such that $BC_0: AC_0 = BA_0: CA_0 = 3$. Prove that the distances from $M$ to $C_0$ and $A_0$ are equal.

2019 Sharygin Geometry Olympiad Finals 8.3 

Construct a regular triangle using a plywood square.  (You can draw a line through  pairs of points lying on the distance less than the side of the square, construct a perpendicular from a point to the line the distance between them does not exceed the side of the square, and measure segments on the constructed lines equal to the side or to the diagonal of the square)

2019 Sharygin Geometry Olympiad Finals 8.4 

Let $O, H$ be the orthocenter and circumcenter of of an acute-angled triangke $ABC$ with $AB<AC$.Let $K$ be the midpoint of $AH$.The line through $K$ perpendicular to $OK$ meet $AB$ and the tangent to  the circumcircle at $A$ at $X$ and $Y$ respectively. Prove that $\angle XOY=\angle AOB$

2019 Sharygin Geometry Olympiad Finals 8.5

A triangle having one angle equal to 45◦ is drawn on the chequered paper
(see.fig.). Find the values of its remaining angles.

2019 Sharygin Geometry Olympiad Finals 8.6 

What is the least positive integer $k$ such that, in every convex 1001-gon, the sum of any k diagonals is greater than or equal to the sum of the remaining diagonals?

2019 Sharygin Geometry Olympiad Finals 8.7

A point $H$ lies on the side $AB$ of regular polygon $ABCDE$. A circle with center $H$ and radius $HE$ meets the segments $DE$ and $CD$ at points $G$ and $F$ respectively. It is known that $DG=AH$. Prove that $CF=AH$.

2019 Sharygin Geometry Olympiad Finals 8.8 

Let points $M$ and $N$ lie on sides $AB$ and $BC$ of triangle $ABC$ in such a way that $MN||AC$. Points $M'$ and $N'$ are the reflections of $M$ and $N$ about $BC$ and $AB$ respectively. Let $M'A$ meet $BC$ at $X$, and let $N'C$ meet $AB$ at $Y$. Prove that $A,C,X,Y$ are concyclic.

grade 9

2019 Sharygin Geometry Olympiad Finals 9.1 

A triangle $OAB$ with $\angle A=90^{\circ}$ lies inside another triangle with vertex $O$. The altitude of $OAB$ from $A$ until it meets the side of angle $O$ at $M$. The distances from $M$ and $B$ to the second side of angle $O$ are $2$ and $1$ respectively. Find the length of $OA$.

2019 Sharygin Geometry Olympiad Finals 9.2

Let $P$ be a point on the circumcircle of triangle $ABC$. Let $A_1$ be the reflection of the orthocenter of triangle $PBC$ about the reflection of the perpendicular bisector of $BC$. Points $B_1$ and $C_1$ are defined similarly. Prove that $A_1,B_1,C_1$ are collinear.

2019 Sharygin Geometry Olympiad Finals 9.3 

Let $ABCD$ be a cyclic quadrilateral such that $AD=BD=AC$. A point $P$ moves along the circumcircle $\omega$ of triangle $ABCD$. The lined $AP$ and $DP$ meet the lines $CD$ and $AB$ at points $E$ and $F$ respectively. The lines $BE$ and $CF$ meet point $Q$. Find the locus of $Q$.

2019 Sharygin Geometry Olympiad Finals 9.4

A ship tries to land in the fog. The crew does not know the direction to the land. They see a lighthouse on a little island, and they understand that the distance to the lighthouse does not exceed 10 km (the exact distance is not known). The distance from the lighthouse to the land equals 10 km. The lighthouse is surrounded by reefs, hence the ship cannot approach it. Can the ship land having sailed the distance not greater than 75 km?
(The waterside is a straight line, the trajectory has to be given before the beginning of the motion, after that the autopilot navigates the ship.)

2019 Sharygin Geometry Olympiad Finals 9.5 

Let $R$ be the circumradius of a circumscribed quadrilateral $ABCD$. Let $h_1$ and $h_2$ be the altitudes from $A$ to $BC$ and $CD$ respectively. Similarly $h_3$ and $h_4$ are the altitudes from $C$ to $AB$ and $AD$. Prove that $$\frac {h_1+h_2- 2R}{h_1h_2}=\frac {h_3+h_4-2R}{h_3h_4}$$  

2019 Sharygin Geometry Olympiad Finals 9.6 

A non-convex polygon has the property that every three consecutive its vertices from a right-angled triangle. Is it true that this polygon has always an angle equal to $90^{\circ}$ or to $270^{\circ}$?

2019 Sharygin Geometry Olympiad Finals 9.7 

Let the incircle $\omega$ of $\triangle ABC$ touch $AC$ and $AB$ at points $E$ and $F$ respectively. Points $X$, $Y$ of $\omega$ are such that $\angle BXC=\angle BYC=90^{\circ}$. Prove that $EF$ and $XY$ meet on the medial line of $ABC$.

2019 Sharygin Geometry Olympiad Finals 9.8

A hexagon $A_1A_2A_3A_4A_5A_6$ has no four concyclic vertices, and its diagonals $A_1A_4$, $A_2A_5$ and $A_3A_6$  concur. Let $l_i$ be the radical axis of circles $A_iA_{i+1}A_{i-2}$ and $A_iA_{i-1}A_{i+2}$ (the points $A_i$ and $A_{i+6}$ coincide). Prove that $l_i, i=1,\cdots,6$, concur.

grade 10

2019 Sharygin Geometry Olympiad Finals 10.1

Given a triangle $ABC$ with $\angle A = 45^\circ$. Let $A'$ be the antipode of $A$ in the circumcircle of $ABC$. Points $E$ and $F$ on segments $AB$ and $AC$ respectively are such that $A'B = BE$, $A'C = CF$. Let $K$ be the second intersection of circumcircles of triangles $AEF$ and $ABC$. Prove that $EF$ bisects $A'K$.

2019 Sharygin Geometry Olympiad Finals 10.2

Let $A_1$, $B_1$, $C_1$ be the midpoints of sides $BC$, $AC$ and $AB$ of triangle $ABC$, $AK$ be the altitude from $A$, and $L$ be the tangency point of the incircle $\gamma$ with $BC$. Let the circumcircles of triangles $LKB_1$ and $A_1LC_1$ meet $B_1C_1$ for the second time at points $X$ and $Y$ respectively, and $\gamma$ meet this line at points $Z$ and $T$. Prove that $XZ = YT$.

2019 Sharygin Geometry Olympiad Finals 10.3

Let $P$ and $Q$ be isogonal conjugates inside triangle $ABC$. Let $\omega$ be the circumcircle of $ABC$. Let $A_1$ be a point on arc $BC$ of $\omega$ satisfying $\angle BA_1P = \angle CA_1Q$. Points $B_1$ and $C_1$ are defined similarly. Prove that $AA_1$, $BB_1$, $CC_1$ are concurrent.

2019 Sharygin Geometry Olympiad Finals 10.4 

Prove that the sum of two nagelians is greater than the semiperimeter of a triangle. 
(The nagelian is the segment between the vertex of a triangle and the tangency point of the opposite side with the correspondent excircle.)

2019 Sharygin Geometry Olympiad Finals 10.5

Let $AA_1, BB_1, CC_1$ be the altitudes of triangle $ABC$, and $A0, C0$ be the common points of the circumcircle of triangle $A_1BC_1$ with the lines $A_1B_1$ and $C_1B_1$ respectively. Prove that $AA_0$ and $CC_0$ meet on the median of ABC or are parallel to it.

2019 Sharygin Geometry Olympiad Finals 10.6

Let $AK$ and $AT$ be the bisector and the median of an acute-angled triangle $ABC$ with $AC > AB$. The line $AT$ meets the circumcircle of $ABC$ at point $D$. Point $F$ is the reflection of $K$ about $T$. If the angles of $ABC$ are known, find the value of angle $FDA$.

2019 Sharygin Geometry Olympiad Finals 10.7 

Let $P$ be an arbitrary point on side $BC$ of triangle $ABC$. Let $K$ be the incenter of triangle $PAB$. Let the incircle of triangle $PAC$ touch $BC$ at $F$. Point $G$ on $CK$ is such that $FG // PK$. Find the locus of $G$.

2019 Sharygin Geometry Olympiad Finals 10.8 

Several points and planes are given in the space. It is known that for any two of given points there exactly two planes containing them, and each given plane contains at least four of given points. Is it true that all given points are collinear?

2019-2020 First Round

2020 Sharygin Geometry Olympiad First Round p1 grade 8
Let $ABC$ be a triangle with $\angle C=90^\circ$, and $A_0$, $B_0$, $C_0$ be the mid-points of sides $BC$, $CA$, $AB$ respectively. Two regular triangles $AB_0C_1$ and $BA_0C_2$ are constructed outside $ABC$. Find the angle $C_0C_1C_2$.

2020 Sharygin Geometry Olympiad First Round p2 grade 8

Let $ABCD$ be a cyclic quadrilateral. A circle passing through $A$ and $B$ meets $AC$ and $BD$ at points $E$ and $F$ respectively. The lines $AF$ and $BC$ meet at point $P$, and the lines $BE$ and $AD$ meet at point $Q$. Prove that $PQ$ is parallel to $CD$.

2020 Sharygin Geometry Olympiad First Round p3 grade 8

Let $ABC$ be a triangle with $\angle C=90^\circ$, and $D$ be a point outside $ABC$, such that $\angle ADC=\angle BAC$. The segments $CD$ and $AB$ meet at point $E$. It is known that the distance from $E$ to $AC$ is equal to the circumradius of triangle $ADE$. Find the angles of triangle $ABC$.

2020 Sharygin Geometry Olympiad First Round p4 grade 8

Let $ABCD$ be an isosceles trapezoid with bases $AB$ and $CD$. Prove that the centroid of triangle $ABD$ lies on $CF$ where $F$ is the projection of $D$ to $AB$.

2020 Sharygin Geometry Olympiad First Round p5 grades 8-9  

Let $BB_1$, $CC_1$ be the altitudes of triangle $ABC$, and $AD$ be the diameter of its circumcircle. The lines $BB_1$ and $DC_1$ meet at point $E$, the lines $CC_1$ and $DB_1$ meet at point $F$. Prove that $\angle CAE = \angle BAF$.

2020 Sharygin Geometry Olympiad First Round p6 grades 8-9 

Circles $\omega_1$ and $\omega_2$ meet at point $P,Q$. Let $O$ be the common point of external tangents of $\omega_1$ and $\omega_2$. A line passing through $O$ meets $\omega_1$ and $\omega_2$ at points $A,B$ located on the same side with respect to line segment $PQ$.The line $PA$ meets $\omega_2$ for the second time at $C$ and the line $QB$ meets $\omega_1$ for the second time at $D$. Prove that $O-C-D$ are collinear.

2020 Sharygin Geometry Olympiad First Round p7 grades 8-9  

Prove that the medial lines of triangle $ABC$ meets the sides of triangle formed by its excenters at six concyclic points.

2020 Sharygin Geometry Olympiad First Round p8 grades 8-9  

Two circles meeting at points $P$ and $R$ are given. Let $\ell_1$, $\ell_2$ be two lines passing through $P$. The line $\ell_1$ meets the circles for the second time at points $A_1$ and $B_1$. The tangents at these points to the circumcircle of triangle $A_1RB_1$ meet at point $C_1$. The line $C_1R$ meets $A_1B_1$ at point $D_1$. Points $A_2$, $B_2, C_2, D_2$ are defined similarly. Prove that the circles $D_1D_2P$ and $C_1C_2R$ touch.

2020 Sharygin Geometry Olympiad First Round p9 grades 8-9 

The vertex $A$, center $O$ and Euler line $\ell$ of a triangle $ABC$ is given. It is known that $\ell$ intersects $AB,AC$ at two points equidistant from $A$. Restore the triangle

2020 Sharygin Geometry Olympiad First Round p10 grades 8-9 

Given are a closed broken line $A_1A_2\ldots A_n$ and a circle $\omega$ which touches each of lines $A_1A_2,A_2A_3,\ldots,A_nA_1$. Call the link good, if it touches $\omega$, and bad otherwise (i.e. if the extension of this link touches $\omega$). Prove that the number of bad links is even.

2020 Sharygin Geometry Olympiad First Round p11 grades 8-9 

Let $ABC$ be a triangle with $\angle A=60^{\circ}$, $AD$ be its bisector, and $PDQ$ be a regular triangle with altitude $DA$. The lines $PB$ and $QC$ meet at point $K$. Prove that $AK$ is a symmedian of $ABC$.

2020 Sharygin Geometry Olympiad First Round p12 grades 8-10 

Let $H$ be the orthocenter of a nonisosceles triangle $ABC$. The bisector of angle $BHC$ meets $AB$ and $AC$ at points $P$ and $Q$ respectively. The perpendiculars to $AB$ and $AC$ from $P$ and $Q$ meet at $K$. Prove that $KH$ bisects the segment $BC$.

2020 Sharygin Geometry Olympiad First Round p13 grades 9-11 

Let $I$ be the incenter of triangle $ABC$. The excircle with center $I_A$ touches the side $BC$ at point $A'$. The line $l$ passing through $I$ and perpendicular to $BI$ meets $I_AA'$ at point $K$ lying on the medial line parallel to $BC$. Prove that $\angle B \leq 60^\circ$.

2020 Sharygin Geometry Olympiad First Round p14 grades 9-11 

A non-isosceles triangle is given. Prove that one of the circles touching internally its incircle and circumcircle and externally one of its excircles passes through a vertex of the triangle.

2020 Sharygin Geometry Olympiad First Round p15 grades 9-11 

A circle passing through the vertices $B$ and $D$ of quadrilateral $ABCD$ meets $AB$, $BC$, $CD$, and $DA$ at points $K$, $L$, $M$, and $N$ respectively. A circle passing through $K$ and $M$ meets $AC$ at $P$ and $Q$. Prove that $L$, $N$, $P$, and $Q$ are concyclic.

2020 Sharygin Geometry Olympiad First Round p16 grades 9-11 

Cevians $AP$ and $AQ$ of a triangle $ABC$ are symmetric with respect to its bisector. Let $X$, $Y$ be the projections of $B$ to $AP$ and $AQ$ respectively, and $N$, $M$ be the projections of $C$ to $AP$ and $AQ$ respectively. Prove that $XM$ and $NY$ meet on $BC$.

2020 Sharygin Geometry Olympiad First Round p17 grades 10-11 

Chords $A_1A_2$ and $B_1B_2$ meet at point $D$. Suppose $D'$ is the inversion image of $D$ and the line $A_1B_1$ meets the perpendicular bisector to $DD'$ at a point $C$. Prove that $CD\parallel A_2B_2$.

2020 Sharygin Geometry Olympiad First Round p18 grades 10-11  

Bisectors $AA_1$, $BB_1$, and $CC_1$ of triangle $ABC$ meet at point $I$. The perpendicular bisector to $BB_1$ meets $AA_1,CC_1$ at points $A_0,C_0$ respectively. Prove that the circumcircles of triangles $A_0IC_0$ and $ABC$ touch.

2020 Sharygin Geometry Olympiad First Round p19 grades 10-11  

Quadrilateral $ABCD$ is such that $AB \perp CD$ and $AD \perp BC$. Prove that there exist a point such that the distances from it to the sidelines are proportional to the lengths of the corresponding sides.

2020 Sharygin Geometry Olympiad First Round p20 grades 10-11  

The line touching the incircle of triangle $ABC$ and parallel to $BC$ meets the external bisector of angle $A$ at point $X$. Let $Y$ be the midpoint of arc $BAC$ of the circumcircle. Prove that the angle $XIY$ is right.

2020 Sharygin Geometry Olympiad First Round p21 grades 10-11 
The diagonals of bicentric quadrilateral $ABCD$ meet at point $L$. Given are three segments equal to $AL$, $BL$, $CL$. Restore the quadrilateral using a compass and a ruler.

2020 Sharygin Geometry Olympiad First Round p22 grades 10-11  

Let $\Omega$ be the circumcircle of cyclic quadrilateral $ABCD$. Consider such pairs of points $P$, $Q$ of diagonal $AC$ that the rays $BP$ and $BQ$ are symmetric with respect the bisector of angle $B$. Find the locus of circumcenters of triangles $PDQ$.

2020 Sharygin Geometry Olympiad First Round p23 grades 10-11  

A non-self-intersecting polygon is nearly convex if precisely one of its interior angles is greater than $180^\circ$.One million distinct points lie in the plane in such a way that no three of them are collinear. We would like to construct a nearly convex one-million-gon whose vertices are precisely the one million given points. Is it possible that there exist precisely ten such polygons?

2020 Sharygin Geometry Olympiad First Round p24 grade 11

Let $I$ be the incenter of a tetrahedron $ABCD$, and $J$ be the center of the exsphere touching the face $BCD$ containing three remaining faces (outside these faces). The segment $IJ$ meets the circumsphere of the tetrahedron at point $K$. Which of two segments $IJ$ and $JK$ is longer?

2020-2021 First Round

2021 Sharygin Geometry Olympiad First Round p1 grade 8

Let $ABC$ be a triangle with $\angle C=90^\circ$. A line joining the midpoint of its altitude $CH$ and the vertex $A$ meets $CB$ at point $K$. Let $L$ be the midpoint of $BC$ ,and $T$ be a point of segment $AB$ such that $\angle ATK=\angle LTB$. It is known that $BC=1$. Find the perimeter of triangle $KTL$.

2021 Sharygin Geometry Olympiad First Round p2 grade 8

A perpendicular bisector to the side $AC$ of triangle $ABC$ meets $BC,AB$ at points $A_1$ and $C_1$ respectively. Points $O,O_1$ are the circumcenters of triangles $ABC$ and $A_1BC_1$ respectively. Prove that $C_1O_1\perp AO$.

2021 Sharygin Geometry Olympiad First Round p3 grade 8

Altitudes $AA_1,CC_1$ of acute-angles $ABC$ meet at point $H$ ; $B_0$ is the midpoint of $AC$. A line passing through $B$ and parallel to $AC$ meets $B_0A_1 , B_0C_1$ at points $A',C'$ respectively. Prove that $AA',CC'$ and $BH$ concur.

2021 Sharygin Geometry Olympiad First Round p4 grade 8

Let $ABCD$ be a square with center $O$ , and $P$ be a point on the minor arc $CD$ of its circumcircle. The tangents from $P$ to the incircle of the square meet $CD$ at points $M$ and $N$. The lines $PM$ and $PN$ meet segments $BC$ and $AD$ respectively at points $Q$ and $R$. Prove that the median of triangle $OMN$ from $O$ is perpendicular to the segment $QR$ and equals to its half.

2021 Sharygin Geometry Olympiad First Round p5 grades 8-9

Five points are given in the plane. Find the maximum number of similar triangles whose vertices are among those five points.

2021 Sharygin Geometry Olympiad First Round p6 grades 8-9

Three circles $\Gamma_1,\Gamma_2,\Gamma_3$ are inscribed into an angle(the radius of $\Gamma_1$ is the minimal, and the radius of $\Gamma_3$ is the maximal) in such a way that $\Gamma_2$ touches $\Gamma_1$ and $\Gamma_3$ at points $A$ and $B$ respectively. Let $\ell$ be a tangent to $A$ to $\Gamma_1$. Consider circles $\omega$ touching $\Gamma_1$ and $\ell$. Find the locus of meeting points of common internal tangents to $\omega$ and $\Gamma_3$.

2021 Sharygin Geometry Olympiad First Round p7 grades 8-9

The incircle of triangle $ABC$ centered at $I$ touches $CA,AB$ at points $E,F$ respectively. Let points $M,N$ of line $EF$ be such that $CM=CE$ and $BN=BF$. Lines $BM$ and $CN$ meet at point $P$. Prove that $PI$ bisects segment $MN$.

2021 Sharygin Geometry Olympiad First Round p8 grades 8-9

Let $ABC$ be an isosceles triangle ($AB=BC$) and $\ell$ be a ray from $B$. Points $P$ and $Q$ of $\ell$ lie inside the triangle in such a way that $\angle BAP=\angle QCA$. Prove that $\angle PAQ=\angle PCQ$.

2021 Sharygin Geometry Olympiad First Round p9 grades 8-9

Points $E$ and $F$ lying on sides $BC$ and $AD$ respectively of a parallelogram $ABCD$ are such that $EF=ED=DC$. Let $M$ be the midpoint of $BE$ and $MD$ meet $EF$ at $G$. Prove that $\angle EAC=\angle GBD$.

2021 Sharygin Geometry Olympiad First Round p10 grades 8-9

Prove that two isotomic lines of a triangle cannot meet inside its medial triangle.

(Two lines are isotomic lines of triangle $ABC$ if their common points with $BC, CA, AB$ are symmetric with respect to the midpoints of the corresponding sides.)

2021 Sharygin Geometry Olympiad First Round p11 grades 8-9

The midpoints of four sides of a cyclic pentagon were marked, after this the pentagon was erased. Restore it.

2021 Sharygin Geometry Olympiad First Round p12 grades 8-10

Suppose we have ten coins with radii $1, 2, 3, \ldots , 10$ cm. We can put two of them on the table in such a way that they touch each other, after that we can add the coins in such a way that each new coin touches at least two of previous ones. The new coin cannot cover a previous one. Can we put several coins in such a way that the centers of some three coins are collinear?

2021 Sharygin Geometry Olympiad First Round p13 grades 9-11

In triangle $ABC$ with circumcircle $\Omega$ and incenter $I$, point $M$ bisects arc $BAC$ and line $\overline{AI}$ meets $\Omega$ at $N\ne A$. The excircle opposite to $A$ touches $\overline{BC}$ at point $E$. Point $Q\ne I$ on the circumcircle of $\triangle MIN$ is such that $\overline{QI}\parallel\overline{BC}$. Prove that the lines $\overline{AE}$ and $\overline{QN}$ meet on $\Omega$.

2021 Sharygin Geometry Olympiad First Round p14 grades 9-11

Let $\gamma_A, \gamma_B, \gamma_C$ be excircles of triangle $ABC$, touching the sides $BC$, $CA$, $AB$ respectively. Let $l_A$ denote the common external tangent to $\gamma_B$ and $\gamma_C$ distinct from $BC$. Define $l_B, l_C$ similarly. The tangent from a point $P$ of $l_A$ to $\gamma_B$ distinct from $l_A$ meets $l_C$ at point $X$. Similarly the tangent from $P$ to $\gamma_C$ meets $l_B$ at $Y$. Prove that $XY$ touches $\gamma_A$.

2021 Sharygin Geometry Olympiad First Round p15 grades 9-11

Let $APBCQ$ be a cyclic pentagon. A point $M$ inside triangle $ABC$ is such that $\angle MAB = \angle MCA$, $\angle MAC = \angle MBA$ and $\angle PMB = \angle QMC = 90^\circ$. Prove that $AM$, $BP$, and $CQ$ concur.

2021 Sharygin Geometry Olympiad First Round p16 grades 9-11

Let circles $\Omega$ and $\omega$ touch internally at point $A$. A chord $BC$ of $\Omega$ touches $\omega$ at point $K$. Let $O$ be the center of $\omega$. Prove that the circle $BOC$ bisects segment $AK$.

2021 Sharygin Geometry Olympiad First Round p17 grades 9-11

Let $ABC$ be an acute-angled triangle. Points $A_0$ and $C_0$ are the midpoints of minor arcs $BC$ and $AB$ respectively. A circle passing though $A_0$ and $C_0$ meet $AB$ and $BC$ at points $P$ and $S$ , $Q$ and $R$ respectively (all these points are distinct). It is known that $PQ\parallel AC$. Prove that $A_0P+C_0S=C_0Q+A_0R$.

2021 Sharygin Geometry Olympiad First Round p18 grades 10-11

Let $ABC$ be a scalene triangle, $AM$ be the median through $A$, and $\omega$ be the incircle. Let $\omega$ touch $BC$ at point $T$ and segment $AT$ meet $\omega$ for the second time at point $S$. Let $\delta$ be the triangle formed by lines $AM$ and $BC$ and the tangent to $\omega$ at $S$. Prove that the incircle of triangle $\delta$ is tangent to the circumcircle of triangle $ABC$.

2021 Sharygin Geometry Olympiad First Round p19 grades 10-11

A point $P$ lies inside a convex quadrilateral $ABCD$. Common internal tangents to the incircles of triangles $PAB$ and $PCD$ meet at point $Q$, and common internal tangents to the incircles of $PBC,PAD$ meet at point $R$. Prove that $P,Q,R$ are collinear.

2021 Sharygin Geometry Olympiad First Round p20 grades 10-11

The mapping $f$ assigns a circle to every triangle in the plane so that the following conditions hold. (We consider all nondegenerate triangles and circles of nonzero radius.)

(a) Let $\sigma$ be any similarity in the plane and let $\sigma$ map triangle $\Delta_1$ onto triangle $\Delta_2$. Then $\sigma$ also maps circle $f(\Delta_1)$ onto circle $f(\Delta_2)$.

(b) Let $A,B,C$ and $D$ be any four points in general position. Then circles $f(ABC),f(BCD),f(CDA)$ and $f(DAB)$ have a common point.

Prove that for any triangle $\Delta$, the circle $f(\Delta)$ is the Euler circle of $\Delta$.

2021 Sharygin Geometry Olympiad First Round p21 grades 10-11

A trapezoid $ABCD$ is bicentral. The vertex $A$, the incenter $I$, the circumcircle $\omega$ and its center $O$ are given and the trapezoid is erased. Restore it using only a ruler.

2021 Sharygin Geometry Olympiad First Round p22 grades 10-11

A convex polyhedron and a point $K$ outside it are given. For each point $M$ of a polyhedron construct a ball with diameter $MK$. Prove that there exists a unique point on a polyhedron which belongs to all such balls.

2021 Sharygin Geometry Olympiad First Round p23 grades 10-11

Six points in general position are given in the space. For each two of them color red the common points (if they exist) of the segment between these points and the surface of the tetrahedron formed by four remaining points. Prove that the number of red points is even.

2021 Sharygin Geometry Olympiad First Round p24 grade 11

A truncated trigonal pyramid is circumscribed around a sphere touching its bases at points $T_1, T_2$. Let $h$ be the altitude of the pyramid, $R_1, R_2$ be the circumradii of its bases, and $O_1, O_2$ be the circumcenters of the bases. Prove that $$R_1R_2h ^ 2 = (R_1 ^ 2-O_1T_1 ^ 2) (R_2 ^ 2-O_2T_2 ^ 2).$$

2020 - 2021 Final Round

grade 8

2021 Sharygin Geometry Olympiad Finals 8.1

Let $ABCD$ be a convex quadrilateral. The circumcenter and the incenter of triangle $ABC$ coincide with the incenter and the circumcenter of triangle $ADC$ respectively. It is known that $AB = 1$. Find the remaining sidelengths and the angles of $ABCD$.

2021 Sharygin Geometry Olympiad Finals 8.2 

Three parallel lines $\ell_a, \ell_b, \ell_c$ pass tlirough the vertices of triangle $ABC$. A line $a$ is the reflection of altitude $AH_a$ about $\ell_a$. Lines $b, c$ are defined similarly. Prove that $a, b, c$ are concurrent.

2021 Sharygin Geometry Olympiad Finals 8.3

Three cockroaches run along a circle in the same direction. They start simultaneously from a point $S$. Cockroach $A$ runs twice as slow than $B$, and tlưee times as slow than $C$. Points $X, Y$ on segment $SC$ are such that $SX = XY =YC$. The lines $AX$ and $BY$ meet at point $Z$. Find the locus of centroids of triangles $ZAB$.

2021 Sharygin Geometry Olympiad Finals 8.4 

Let $A_1$ and $C_1$ be the feet of altitudes $AH$ and $CH$ of an acute-angled triangle $ABC$. Points $A_2$ and $C_2$ are the reflections of $A_1$ and $C_1$ about $AC$. Prove that the distance between the circumcenters of triangles $C_2HA_1$ and $C_1HA_2$ equals $AC$.

2021 Sharygin Geometry Olympiad Finals 8.5

Points $A_1,A_2,A_3,A_4$ are not concyclic, the same for points $B_1,B_2,B_3,B_4$. For all $i, j, k$ the circumradii of triangles $A_iA_jA_k$ and $B_iB_jB_k$ are equal. Can we assert that $A_iA_j=B_iB_j$ for all $i, j$'?

2021 Sharygin Geometry Olympiad Finals 8.6 

Let $ABC$ be an acute-angled triangle. Point $P$ is such that $AP = AB$ and $PB\parallel AC$. Point $Q$ is such that $AQ = AC$ and $CQ\parallel AB$. Segments $CP$ and $BQ$ meet at point $X$. Prove that the circumcenter of triangle $ABC$ lies on the circle $(PXQ)$.

2021 Sharygin Geometry Olympiad Finals 8.7

Let $ABCDE$ be a convex pentagon such that angles $CAB$, $BCA$, $ECD$, $DEC$ and $AEC$ are equal. Prove that $CE$ bisects $BD$.

2021 Sharygin Geometry Olympiad Finals 8.8

Does there exist a convex polygon such that all its sidelengths are equal and all triangle formed by its vertices are obtuse-angled?

grade 9

2021 Sharygin Geometry Olympiad Finals 9.1

Three cevians concur at a point lying inside a triangle. The feet of these cevians divide the sides into six segments, and the lengths of these segments form (in some order) a geometric progression. Prove that the lengths of the cevians also form a geometric progression.

2021 Sharygin Geometry Olympiad Finals 9.2 

A cyclic pentagon is given. Prove that the ratio of its area to the sum of the diagonals is not greater than the quarter of the circumradius.

2021 Sharygin Geometry Olympiad Finals 9.3

Let $ABC$ be an acute-angled scalene triangle and $T$ be a point inside it such that $\angle ATB  = \angle BTC = 120^o$. A circle centered at point $E$ passes through the midpoints of the sides of $ABC$. For $B, T, E$ collinear, find angle $ABC$.

2021 Sharygin Geometry Olympiad Finals 9.4

Define the distance between two triangles to be the closest distance between two vertices, one from each triangle. Is it possible to draw five triangles in the plane such that for any two of them, their distance equals the sum of their circumradii?

2021 Sharygin Geometry Olympiad Finals 9.5 

Let $O$ be the clrcumcenter of triangle $ABC$. Points $X$ and $Y$ on side $BC$ are such that $AX = BX$ and $AY = CY$. Prove that the circumcircle of triangle $AXY$ passes through the circumceuters of triangles $AOB$ and $AOC$.

2021 Sharygin Geometry Olympiad Finals 9.6

The diagonals of trapezoid $ABCD$ ($BC\parallel AD$) meet at point $O$. Points $M$ and $N$ lie on the segments $BC$ and $AD$ respectively. The tangent to the circle $AMC$ at $C$ meets the ray $NB$ at point $P$; the tangent to the circle $BND$ at $D$ meets the ray $MA$ at point $R$. Prove that $\angle BOP =\angle AOR$.

2021 Sharygin Geometry Olympiad Finals 9.7 

Three sidelines of on acute-angled triangle are drawn on the plane. Fyodor wants to draw the altitudes of this triangle using a ruler and a compass. Ivan obstructs him using an eraser. For each move Fyodor may draw one line through two markeed points or one circle centered at a marked point and passing through another marked point. After this Fyodor may mark an arbitrary number of points (the common points of drawn lines, arbitrary points on the drawn lines or arbitrary points on the plane). For each move Ivan erases at most three of marked point. (Fyodor may not use the erased points in his constructions but he may mark them for the second time). They move by turns, Fydors begins. Initially no points are marked. Can Fyodor draw the altitudes?

2021 Sharygin Geometry Olympiad Finals 9.8

A quadrilateral $ABCD$ is circumscribed around a circle $\omega$ centered at $I$. Lines $AC$ and $BD$ meet at point $P$, lines $AB$ and $CD$ meet at point $£$, lines $AD$ and $BC$ meet at point $F$. Point $K$ on the circumcircle of triangle $E1F$ is such that $\angle IKP = 90^o$. The ray $PK$ meets $\omega$ at point $Q$. Prove that the circumcircle of triangle $EQF$ touches $\omega$.

grades 10-11

2021 Sharygin Geometry Olympiad Finals 10-11.1 

Let $CH$ be an altitude of right-angled triangle $ABC$ ($\angle C = 90^o$), $HA_1$, $HB_1$ be the bisectors of angles $CHB$, $AHC$ respectively, and $E, F$ be the midpoints of $HB_1$ and $HA_1$ respectively. Prove that the lines $AE$ and $BF$ meet on the bisector of angle $ACB$.

2021 Sharygin Geometry Olympiad Finals 10-11.2 

Let A$BC$ be a scalene triangle, and $A_o$, $B_o,$ $C_o$ be the midpoints of $BC$, $CA$, $AB$ respectively. The bisector of angle $C$ meets $A_oCo$ and $B_oC_o$ at points $B_1$ and $A_1$ respectively. Prove that the lines $AB_1$, $BA_1$ and $A_oB_o$ concur.

2021 Sharygin Geometry Olympiad Finals 10-11.3

The bisector of angle $A$ of triangle $ABC$ ($AB > AC$) meets its circumcircle at point $P$. The perpendicular to $AC$ from $C$ meets the bisector of angle $A$ at point $K$. A cừcle with center $P$ and radius $PK$ meets the minor arc $PA$ of the circumcircle at point $D$. Prove that the quadrilateral $ABDC$ is circumscribed.

2021 Sharygin Geometry Olympiad Finals 10-11.4

Can a triangle be a development of a quadrangular pyramid?

2021 Sharygin Geometry Olympiad Finals 10-11.5

A secant meets one circle at points $A_1$, $B_1$։, this secant meets a second circle at points $A_2$, $B_2$. Another secant meets the first circle at points $C_1$, $D_1$ and meets the second circle at points $C_2$, $D_2$. Prove that point $A_1C_1 \cap  B_2D_2$, $A_1C_1 \cap  A_2C_2$, $A_2C_2  \cap B_1D_1$, $B_2D_2 \cap B_1D_1$ lie on a circle coaxial with two given circles.

2021 Sharygin Geometry Olympiad Finals 10-11.6 

The lateral sidelines $AB$ and $CD$ of trapezoid $ABCD$ meet at point $S$. The bisector of angle $ASC$ meets the bases of the trapezoid at points $K$ and $L$ ($K$ lies inside segment $SL$). Point $X$ is chosen on segment $SK$, and point $Y$ is selected on the extension of $SL$ beyond $L$ such a way that $\angle AXC - \angle AYC = \angle ASC$. Prove that $\angle BXD - \angle BYD = \angle  BSD$.

2021 Sharygin Geometry Olympiad Finals 10-11.7 

Let $I$ be the incenter of a right-angled triangle $ABC$, and $M$ be the midpoint of hypothenuse $AB$. The tangent to the circumcircle of $ABC$ at $C$ meets the line passing through $I$ and parallel to $AB$ at point $P$. Let $H$ be the orthocenter of triangle $PAB$. Prove that lines $CH$ and $PM$ meet at the incircle of triangle $ABC$.

2021 Sharygin Geometry Olympiad Finals 10-11.8

On the attraction "Merry parking", the auto has only two position* of a steering wheel: "right", and "strongly right". So the auto can move along an arc with radius $r_1$ or $r_1$. The auto started from a point $A$ to the Nord, it covered the distance $\ell$ and rotated to the angle $a < 2\pi$. Find the locus of its possible endpoints.

2021-2022 Final Round

2022 Sharygin Geometry Olympiad First Round p1 grade 8

Let $O$ and $H$ be the circumcenter and the orthocenter respectively of triangle $ABC$. Itis known that $BH$ is the bisector of angle $ABO$. The line passing through $O$ and parallel to $AB$ meets $AC$ at $K$. Prove that $AH = AK$

2022 Sharygin Geometry Olympiad First Round p2 grade 8

Let $ABCD$ be a curcumscribed quadrilateral with incenter $I$, and let $O_{1}, O_{2}$ be the circumcenters of triangles $AID$ and $CID$. Prove that the circumcenter of triangle $O_{1}IO_{2}$ lies on the bisector of angle $ABC$

2022 Sharygin Geometry Olympiad First Round p3 grade 8

Let $CD$ be an altitude of right-angled triangle $ABC$ with $\angle C = 90$. Regular triangles$AED$ and $CFD$ are such that $E$ lies on the same side from $AB$ as $C$, and $F$ lies on the same side from $CD$ as $B$. The line $EF$ meets $AC$ at $L$. Prove that $FL = CL + LD$

2022 Sharygin Geometry Olympiad First Round p4 grade 8

Let $AA_1$, $BB_1$, $CC_1$ be the altitudes of acute angled triangle $ABC$. $A_2$ be the touching point of the incircle of triangle $AB_1C_1$ with $B_1C_1$, points $B_2$, $C_2$ be defined similarly. Prove that the lines $A_1A_2$, $B_1B_2$, $C_1C_2$ concur.

2022 Sharygin Geometry Olympiad First Round p5 grade 8

Let the diagonals of cyclic quadrilateral $ABCD$ meet at point $P$. The line passing through $P$ and perpendicular to $PD$ meets $AD$ at point $D_1$, a point $A_1$ is defined similarly. Prove that the tangent at $P$ to the circumcircle of triangle $D_1PA_1$ is parallel to $BC$.

2022 Sharygin Geometry Olympiad First Round p6  grades 8-9

The incircle and the excircle of triangle $ABC$ touch the side $AC$ at points $P$ and $Q$ respectively. The lines $BP$ and $BQ$ meet the circumcircle of triangle $ABC$ for the second time at points $P'$ and $Q'$ respectively. Prove that $$PP' > QQ'$$

2022 Sharygin Geometry Olympiad First Round p7  grades 8-9

A square with center $F$ was constructed on the side $AC$ of triangle $ABC$ outside it. After this, everything was erased except $F$ and the midpoints $N,K$ of sides $BC,AB$.

Restore the triangle.

2022 Sharygin Geometry Olympiad First Round p8  grades 8-9

Points $P,Q,R$ lie on the sides $AB,BC,CA$ of triangle $ABC$ in such a way that $AP=PR, CQ=QR$. Let $H$ be the orthocenter of triangle $PQR$, and $O$ be the circumcenter of triangle $ABC$. Prove that $$OH||AC$$ .

2022 Sharygin Geometry Olympiad First Round p9  grades 8-9

The sides $AB, BC, CD$ and $DA$ of quadrilateral $ABCD$ touch a circle with center $I$ at points $K, L, M$ and $N$ respectively. Let $P$ be an arbitrary point of line $AI$. Let $PK$ meet $BI$ at point $Q, QL$ meet $CI$ at point $R$, and $RM$ meet $DI$ at point $S$. Prove that $P,N$ and $S$ are collinear.

2022 Sharygin Geometry Olympiad First Round p10  grades 8-9

Let $\omega_1$ be the circumcircle of triangle $ABC$ and $O$ be its circumcenter. A circle $\omega_2$ touches the sides $AB, AC$, and touches the arc $BC$ of $\omega_1$ at point $K$. Let $I$ be the incenter of $ABC$. Prove that the line $OI$ contains the symmedian of triangle $AIK$.

2022 Sharygin Geometry Olympiad First Round p11  grades 8-10

Let $ABC$ be a triangle with $\angle A=60^o$ and $T$ be a point such that $\angle ATB=\angle BTC=\angle ATC$. A circle passing through $B,C$ and $T$ meets $AB$ and $AC$ for the second time at points $K$ and $L$.Prove that the distances from $K$ and $L$ to $AT$ are equal.

2022 Sharygin Geometry Olympiad First Round p12  grades 8-11

Let $K$, $L$, $M$, $N$ be the midpoints of sides $BC$, $CD$, $DA$, $AB$ respectively of a convex quadrilateral $ABCD$. The common points of segments $AK$, $BL$, $CM$, $DN$ divide each of them into three parts. It is known that the ratio of the length of the medial part to the length of the whole segment is the same for all segments. Does this yield that $ABCD$ is a parallelogram?

2022 Sharygin Geometry Olympiad First Round p13 grades 8-11

Eight points in a general position are given in the plane. The areas of all $56$ triangles with vertices at these points are written in a row. Prove that it is possible to insert the symbols "$+$" and "$-$" between them in such a way that the obtained sum is equal to zero.

2022 Sharygin Geometry Olympiad First Round p14

A triangle $ABC$ is given. Let $C'$ and $C'_{a}$ be the touching points of sideline $AB$ with the incircle and with the excircle touching the side $BC$. Points $C'_{b}$, $C'_{c}$, $A'$, $A'_{a}$, $A'_{b}$, $A'_{c}$, $B'$, $B'_{a}$, $B'_{b}$, $B'_{c}$ are defined similarly. Consider the lengths of $12$ altitudes of triangles $A'B'C'$, $A'_{a}B'_{a}C'_{a}$, $A'_{b}B'_{b}C'_{b}$, $A'_{c}B'_{c}C'_{c}$.

(a) (8-9) Find the maximal number of different lengths.

(b) (10-11) Find all possible numbers of different lengths.

2022 Sharygin Geometry Olympiad First Round p15  grades 9-11

A line $l$ parallel to the side $BC$ of triangle $ABC$ touches its incircle and meets its circumcircle at points $D$ and $E$. Let $I$ be the incenter of $ABC$. Prove that $AI^2 = AD \cdot AE$.

2022 Sharygin Geometry Olympiad First Round p16  grades 9-11

Let $ABCD$ be a cyclic quadrilateral, $E = AC \cap BD$, $F = AD \cap BC$. The bisectors of angles $AFB$ and $AEB$ meet $CD$ at points $X, Y$ . Prove that $A, B, X, Y$ are concyclic.

2022 Sharygin Geometry Olympiad First Round p17  grades 9-11

Let a point $P$ lie inside a triangle $ABC$. The rays starting at $P$ and crossing the sides $BC$, $AC$, $AB$ under the right angle meet the circumcircle of $ABC$ at $A_{1}$, $B_{1}$, $C_{1}$ respectively. It is known that lines $AA_{1}$, $BB_{1}$, $CC_{1}$ concur at point $Q$. Prove that all such lines $PQ$ concur.

2022 Sharygin Geometry Olympiad First Round p18  grades 10-11

The products of the opposite sidelengths of a cyclic quadrilateral $ABCD$ are

equal. Let $B'$ be the reflection of $B$ about $AC$. Prove that the circle passing through $A,B', D$ touches $AC$

2022 Sharygin Geometry Olympiad First Round p19  grades 10-11

Let $I$ be the incenter of triangle $ABC$, and $K$ be the common point of $BC$ with the external bisector of angle $A$. The line $KI$ meets the external bisectors of angles $B$ and $C$ at points $X$ and $Y$ . Prove that $\angle BAX = \angle CAY$

2022 Sharygin Geometry Olympiad First Round p20  grades 10-11

Let $O$, $I$ be the circumcenter and the incenter of $\triangle ABC$; $R$,$r$ be the circumradius and the inradius; $D$ be the touching point of the incircle with $BC$; and $N$ be an arbitrary point of segment $ID$. The perpendicular to $ID$ at $N$ meets the circumcircle of $ABC$ at points $X$ and $Y$ . Let $O_{1}$ be the circumcircle of $\triangle XIY$.

Find the product $OO_{1}\cdot IN$.

2022 Sharygin Geometry Olympiad First Round p21  grades 10-11

The circumcenter $O$, the incenter $I$, and the midpoint $M$ of a diagonal of a bicentral quadrilateral were marked. After this the quadrilateral was erased. Restore it.

2022 Sharygin Geometry Olympiad First Round p22  grades 10-11

Chords $A_1A_2, A_3A_4, A_5A_6$ of a circle $\Omega$ concur at point $O$. Let $B_i$ be the second common point of $\Omega$ and the circle with diameter $OA_i$ . Prove that chords $B_1B_2, B_3B_4, B_5B_6$ concur.

2022 Sharygin Geometry Olympiad First Round p23  grades 10-11

An ellipse with focus $F$ is given. Two perpendicular lines passing through $F$ meet the ellipse at four points. The tangents to the ellipse at these points form a quadrilateral circumscribed around the ellipse. Prove that this quadrilateral is inscribed into a conic with focus $F$

2022 Sharygin Geometry Olympiad First Round p24  grade 11

Let $OABCDEF$ be a hexagonal pyramid with base $ABCDEF$ circumscribed around a sphere $\omega$. The plane passing through the touching points of $\omega$ with faces $OFA$, $OAB$ and $ABCDEF$ meets $OA$ at point $A_1$, points $B_1$, $C_1$, $D_1$, $E_1$ and $F_1$ are defined similarly. Let $\ell$, $m$ and $n$ be the lines $A_1D_1$, $B_1E_1$ and $C_1F_1$ respectively. It is known that $\ell$ and $m$ are coplanar, also $m$ and $n$ are coplanar. Prove that $\ell$ and $n$ are coplanar.

2021 - 2022 Final Round

2022 Sharygin Geometry Olympiad Finals 8.1

Let $ABCD$ be a convex quadrilateral with $\angle{BAD} = 2\angle{BCD}$ and $AB = AD$. Let $P$ be a point such that $ABCP$ is a parallelogram. Prove that $CP = DP$.

2022 Sharygin Geometry Olympiad Finals 8.2

Let $ABCD$ be a right-angled trapezoid and $M$ be the midpoint of its greater lateral side $CD$. Circumcircles $\omega_{1}$ and $\omega_{2}$ of triangles $BCM$ and $AMD$ meet for the second time at point $E$. Let $ED$ meet $\omega{1}$ at point $F$, and $FB$ meet $AD$ at point $G$. Prove that $GM$ bisects angle $BGD$.

2022 Sharygin Geometry Olympiad Finals 8.3

A circle $\omega$ and a point $P$ not lying on it are given. Let $ABC$ be an arbitrary equilateral triangle inscribed into $\omega$ and A′, B′, C′ be the projections of $P$ to $BC, CA, AB$. Find the locus of centroids of triangles A′B′C′.

2022 Sharygin Geometry Olympiad Finals 8.4

Let $ABCD$ be a cyclic quadrilateral, $O$ be its circumcenter, $P$ be a common points of its diagonals, and $M , N$ be the midpoints of $AB$ and $CD$ respectively. A circle $OPM$ meets for the second time segments $AP$ and $BP$ at points $A_1$ and $B_1$ respectively and a circle $OPN$ meets for the second time segments $CP$ and $DP$ at points $C_1$ and $D_1$ respectively. Prove that the areas of quadrilaterals $AA_1B_1B$ and $CC_1D_1D$ are equal.

2022 Sharygin Geometry Olympiad Finals 8.5

An incircle of triangle $ABC$ touches $AB$, $BC$, $AC$ at points $C_1$, $A_1$,$B_1$ respectively. Let $A'$ be the reflection of $A_1$ about $B_1C_1$, point $C'$ is defined similarly. Lines $A'C_1$ and $C'A_1$ meet at point $D$. Prove that $BD \parallel AC$.

2022 Sharygin Geometry Olympiad Finals 8.6

Two circles meeting at points $A,  B$ and a point $O$ lying outside them are given. Using a compass and a ruler construct a ray with origin $O$ meeting the first circle at point $C$ and the second one at point $D$ in such a way that the ratio $OC : OD$ be maximal.

2022 Sharygin Geometry Olympiad Finals 8.7

Ten points on a plane a such that any four of them lie on the boundary of some square. Is obligatory true that all ten points lie on the boundary of some square?

2022 Sharygin Geometry Olympiad Finals 8.8

An isosceles trapezoid $ABCD$ ($AB = CD$) is given. A point $P$ on its circumcircle is such that segments $CP$ and $AD$ meet at point $Q$. Let $L$ be tha midpoint of$QD$. Prove that the diagonal of the trapezoid is not greater than the sum of distances from the midpoints of the lateral sides to ana arbitrary point of line $PL$.

2022 Sharygin Geometry Olympiad Finals 9.1

Let $BH$ be an altitude of right angled triangle $ABC$($\angle B = 90^o$). An excircle of triangle $ABH$ opposite to $B$ touches $AB$ at point $A_1$; a point $C_1$ is defined similarly. Prove that $AC // A_1C_1$.

2022 Sharygin Geometry Olympiad Finals 9.2

Let circles $s_1$ and $s_2$ meet at points $A$ and $B$. Consider all lines passing through $A$ and meeting the circles for the second time at points $P_1$ and $P_2$ respectively. Construct by a compass and a ruler a line such that $AP_1.AP_2$ is maximal.

2022 Sharygin Geometry Olympiad Finals 9.3

A medial line parallel to the side $AC$ of triangle $ABC$ meets its circumcircle at points at $X$ and $Y$. Let $I$ be the incenter of triangle $ABC$ and $D$ be the midpoint of arc $AC$ not containing $B$.A point $L$ lie on segment $DI$ in such a way that $DL= BI/2$. Prove that $\angle IXL = \angle IYL$.

2022 Sharygin Geometry Olympiad Finals 9.4

Let $ABC$ be an isosceles triangle with $AB = AC$, $P$ be the midpoint of the minor arc $AB$ of its circumcircle, and $Q$ be the midpoint of $AC$. A circumcircle of triangle $APQ$ centered at $O$ meets $AB$ for the second time at point $K$. Prove that lines $PO$ and $KQ$ meet on the bisector of angle $ABC$.

2022 Sharygin Geometry Olympiad Finals 9.5

Chords $AB$ and $CD$ of a circle $\omega$ meet at point $E$ in such a way that $AD = AE = EB$. Let $F$ be a point of segment $CE$ such that $ED = CF$. The bisector of angle $AFC$ meets an arc $DAC$ at point $P$. Prove that $A$, $E$, $F$, and $P$ are concyclic.

2022 Sharygin Geometry Olympiad Finals 9.6

Lateral sidelines $AB$ and $CD$ of a trapezoid $ABCD$ ($AD >BC$) meet at point $P$. Let $Q$ be a point of segment $AD$ such that $BQ = CQ$. Prove that the line passing through the circumcenters of triangles $AQC$ and $BQD$ is perpendicular to $PQ$.

2022 Sharygin Geometry Olympiad Finals 9.7

Let $H$ be the orthocenter of an acute-angled triangle $ABC$. The circumcircle of triangle $AHC$ meets segments $AB$ and $BC$ at points $P$ and $Q$. Lines $PQ$ and $AC$ meet at point $R$. A point $K$ lies on the line $PH$ in such a way that $\angle KAC = 90^{\circ}$. Prove that $KR$ is perpendicular to one of the medians of triangle $ABC$.

2022 Sharygin Geometry Olympiad Finals 9.8

Several circles are drawn on the plane and all points of their intersection or touching are marked. Is it possible that each circle contains exactly five marked points and each point belongs to exactly five circles?

2022 Sharygin Geometry Olympiad Finals 10.1

$A_1A_2A_3A_4$ and $B_1B_2B_3B_4$ are two squares with their vertices arranged clockwise.The perpendicular bisector of segment $A_1B_1,A_2B_2,A_3B_3,A_4B_4$ and the perpendicular bisector of segment $A_2B_2,A_3B_3,A_4B_4,A_1B_1$ intersect at point $P,Q,R,S$ respectively.Show that:$PR\perp QS$.

2022 Sharygin Geometry Olympiad Finals 10.2

Let $ABCD$ be a convex quadrilateral. The common external tangents to circles $(ABC)$ and $(ACD)$ meet at point $E$, the common external tangents to circles $(ABD)$ and $(BCD)$ meet at point $F$. Let $F$ lie on $AC$, prove that $E$ lies on $BD$.

2022 Sharygin Geometry Olympiad Finals 10.3

A line meets a segment $AB$ at point $C$. Which is the maximal number of points $X$ of this line such that one of angles $AXC$ and $BXC$ is equlal to a half of the second one?

2022 Sharygin Geometry Olympiad Finals 10.4

Let $ABCD$ be a convex quadrilateral with $\angle B= \angle D$. Prove that the midpoint of $BD$ lies on the common internal tangent to the incircles of triangles $ABC$ and $ACD$.

2022 Sharygin Geometry Olympiad Finals 10.5

Let$AB$ and $AC$ be the tangents from a point $A$ to a circle $\Omega$. Let $M$ be the midpoint of $BC$ and $P$ be an arbitrary point on this segment. A line $AP$ meets $\Omega$ at points $D$ and $E$. Prove that the common external tangents to circles $MDP$ and $MPE$ meet on the midline of triangle $ABC$.

2022 Sharygin Geometry Olympiad Finals 10.6

Let $O, I$ be the circumcenter and the incenter of triangle $ABC$, $P$ be an arbitrary point on segment $OI$, $P_A$, $P_B$, and $P_C$ be the second common points of lines $PA$, $PB$, and $PC$ with the circumcircle of triangle $ABC$. Prove that the bisectors of angles $BP_AC$, $CP_BA$, and $AP_CB$ concur at a point lying on $OI$.

2022 Sharygin Geometry Olympiad Finals 10.7

Several circles are drawn on the plane and all points of their meeting or touching are marked. May be that each circle contains exactly four marked points and exactly four marked points lie on each circle?

2022 Sharygin Geometry Olympiad Finals 10.8

Let $ABCA'B'C'$ be a centrosymmetric octahedron (vertices $A$ and $A'$, $B$ and $B'$, $C$ and $C'$ are opposite) such that the sums of four planar angles equal $240^o$ for each vertex. The Torricelli points $T_1$ and $T_2$ of triangles $ABC$ and $A'BC$ are marked. Prove that the distances from $T_1$ and $T_2$ to $BC$ are equal.

sources:   geometry.ru/olimp/olimpsharygin.php , sasja.shap.homedns.org/Turniry/Shar-eng/