### Sharygin Geometry 2005-19 691p

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

below are 2005  - 2019: First &  Final Round

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 2010 final in English inside aops here
Sharygin aops collection under costruction: 2005, 2006

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

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

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

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.

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?

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.

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?

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?

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

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

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 $PO$.

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

The square was cut into $n$ 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?

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

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.

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 $A_1B_1S_1$.

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

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.

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.

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?

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.

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 $A'B'C'D '$.

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?

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.

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

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

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.

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

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.

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 $OH$.

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 $135^o$

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.

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

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 $CD$.

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)
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 $AA'$.

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.

$A_1, B_1, C_1$ 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 $ABC$.

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.

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 $ABCD$.

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 $BC$.

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 $AMB$.

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

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?

Points $A, B$ move with equal speeds along two equal circles.Prove that the perpendicular bisector of  $AB$ passes through a fixed point.

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 $r$.

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.

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 $AB$.

$L(a)$ 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.

At what $n$ can a regular $n$-gon be cut by disjoint diagonals into $n- 2$ isosceles (including equilateral) triangles?

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.

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 $AB = AC$.

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 $XB$.

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.

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.

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?

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 $XY // A_1B_1$

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.

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 $T$.

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 $T$.

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.

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 $PXY$.

$ABCD$ 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 $NG: GI = 2: 1$.

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.

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 $AB / CD$.

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

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

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

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 $DX = DY$.

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.

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

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 $ABC$.

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.

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.

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.

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?

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.

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.

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.

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.

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.

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.

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)?

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

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.

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

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?

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

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?

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?

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?

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,

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,

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.

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.)

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 $PQ \perp AB$.

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.

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 $\angle PBA = \angle DBC$.

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 $\angle PAC = \angle QAB$.

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 $PQ$.

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

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

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?
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.

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

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

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 $B$.

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

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

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.

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?

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

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 $AE \cdot ED = AF \cdot FB$.

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.

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 $A'B'$.

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

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.

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 $C_0$.

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 $ABC$).

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 $ABC$.

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 $DAB$.

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?

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

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

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

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

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
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
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
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
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

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
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
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
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
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
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
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
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
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
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
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
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
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
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

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

by B.Frenkin
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
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
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
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
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
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
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

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
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
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
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
Can the surface of a regular tetrahedron be glued over with equal regular hexagons?

by N.Avilov
Construct the triangle, given its centroid and the feet of an altitude and a bisector from the same vertex.

by B.Frenkin
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
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
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
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
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
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
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
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
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
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

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.

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?

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

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.

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 $ABC$.

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

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 $ABC$.

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?

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 $k > 0.666n$.

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 $C$.

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 $CD$.

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.

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.

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

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.

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.

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 $ABC$.

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).

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 $n$?

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.

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 $ABCD$.

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

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

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 $\angle BAC$.

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.

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
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
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
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
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

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

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.

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

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

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

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

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

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

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

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
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

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

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

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

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

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

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

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?

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 $OI \perp AB.$

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.

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 $N.$

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.

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 $AMN.$

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 $O_1M=O_2M.$

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 $\angle LI_bI_c.$

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.

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 $\frac{AD}{BD}=\frac{BC}{AC}.$

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 $n.$

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.

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 $AK \cdot CD = KC \cdot AD.$

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.

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) $\frac{A_1N}{BB_1} + \frac{A_1L}{CC_1}=1.$

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 $DE.$

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.

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 $AC.$

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 $XV=YU.$

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 $C.$

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

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.

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.

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 $Y.$

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

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 $A$.

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
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 $AB = CD$.

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.

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.

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

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?

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?

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?

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.)

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
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.

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 $A_0BC_0$.

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 $AMK$.

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.

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.

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 $OJ$.

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 $cos \angle A + cos \angle B = 1$.

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?

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.

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 $OB_1 = OB_2$.

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?

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?

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
(under constuction)

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

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 $BC$.

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 $ABC$.

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?

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.

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.

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

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 $ABC$.

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 $A_1H = C_1H$.

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 $O$.

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 $AC||A'C'$.

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 $\angle AQM = \angle BPM$.

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).

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 $ABC$.

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 $MA$.

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.

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?

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?

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?

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 $IM : AC = IN : BD$.

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.

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.

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.

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.

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
(under constuction)

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

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.

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.

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.

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.

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 $ABC$.

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 $P Q$.

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.

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.

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 $CPB$.

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

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 $IJ$.

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?

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.

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 $AB = EF$.

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?

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 $ABC$.

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.

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.

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 $ABC$

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.

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.

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.

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

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 $AC = 2 MD$

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?

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 $\angle Y B_{1}Z = \angle XB_{1}Z$.

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.

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 $A_{1}C_{1}\parallel AC$.

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 $A_{2}C_{2}$.

In a non-isosceles triangle $ABC$ the bisectors of angles $A$ and $B$ are inversely proportional to the respective sidelengths. Find angle $C$.

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 $ABC$.

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 $ABC$.

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?

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.

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 $PB + QC = PQ$.

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.

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

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.

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 $ACM$.

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.

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.

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 $XZY$.

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 $AB$.

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.

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.

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.

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

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 $\angle AKC$.

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
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
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
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
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
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
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

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
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
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
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
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
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
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
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

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
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
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

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

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

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
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

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

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 $\angle DAC$.

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.

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.

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 $PA$.

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?

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 $BM = CM$.

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 $\angle DBQ = 90^\circ$.

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.

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 $\angle BCA$.

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.

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?

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.

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.

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.

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.

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 $CK // AB$.

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 $\phi$.

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 $\angle BAX = \angle CAY$.

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.

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.

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 $BC$.

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

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

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 $FA = AB$.

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.

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?

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.

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 $C'K = A'K$.

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.

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.

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.

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 $B_1D_1 // AE$

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 $O_1O_2$.

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 $S_{AOB}+ S_{COD} \le 2(S_{AOD}+ S_{BOC})$.

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 $ABC$.

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 $\frac{BK}{CK}=\frac{FS}{ES}$ .
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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 $I_aI_c$.

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.

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?

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 $\angle PA_1Q = 2\angle BAC$.

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.

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.

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.)

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 $O$.

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 $PQ$.

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.

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

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 $M$.

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 $45^\circ$?

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.

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.

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 $ABC$.

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 $Y$.

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 $EF$.

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 $BF\perp KL$.

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 $MQ\parallel AL$.

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.

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 $EF\parallel AB$.

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 $AB$.

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.

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.

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)?

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.

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.

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.

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.

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 $PQ$.

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 $E_1 F_1$.

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

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.

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

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
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
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 $\angle ABM$.

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
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
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
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.

Folklor
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

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

by V. Yasinsky
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
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
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
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
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
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
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

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
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
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)?

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
The altitude from one vertex of a triangle, the bisector from the another 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
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 E. H. Garsia
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
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

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?

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 $ABC$.

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 $AED$.

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.

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 $\angle DAB = 2\angle ADB.$

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.

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 $A' C' // AC$.

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 $FCD$.

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 $A'C'B'$,
- $A'C' + B'C'= AB$.

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

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 $AC$.

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

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 $AC$.

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

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 $ABC$.

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.

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.

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.

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 $LKP$.

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 $LKP$.

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 $LKP$.

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.

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.

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

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

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

by N. Belukhov
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
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
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
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

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
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
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
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
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
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
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
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.

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 $AKB$.

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

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 $ABC$.

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 $A_2B_2C_2$.

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 $\angle MBK = 90$.

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.

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.

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

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

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.

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 $H_2F // BC$.

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 $ABD$.

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 $DCA$.

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 $AC /4$.

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 $n$)?

Let $ABCDE$ be an inscribed pentagon such that  $\angle B + \angle E = \angle C + \angle D$. Prove that $\angle CAD < \pi / 3 < \angle A$.

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 $AB$.

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 $ABC$.

Restore a triangle $ABC$ by vertex $B$, the centroid and the common point of the symmedian from $B$ with the circumcircle.

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.

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.

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 $\angle A$.

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 $60^o$.

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 $ABC$.

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 $BC$.

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 $ABC$.

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 $QC$.

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 $\omega$ .

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 $Q$.

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 $ABC$.

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)

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

An altitude $AH$ of triangle $ABC$ bisects a median $BM$. Prove that the medians of triangle $ABM$ are sidelengths of a right-angled triangle.

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
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
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
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
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
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
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
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 $AG=AB$.

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.

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 $XY$?

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?

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 $RS$.

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 $\angle PMB=\angle QMB$.

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 $\angle PMB=\angle QMB$.

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
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.

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
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 $A'B'C'$.

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
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)

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
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)

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

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

by A.Zaslavsky
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
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
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

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.

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
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
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
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
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
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

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
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
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
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 $IM$.

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.

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

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 $\angle CLH = 180^o - 2\angle C$.

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.

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).

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 $r.$

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 $ML$.

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.

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
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 $PQ \parallel AC$.

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 $A_1OC_1$.

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.

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.

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 $AB$.

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.

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.

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 $AB$.

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.

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 $CI:IW$.

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
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.

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
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 $\alpha$.

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.

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
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 $AB$.

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.

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 $X_A + X_B + X_C + X_D = 0$.

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.

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.

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 $SFC$.

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.

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
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.

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 $AFB$?

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.

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.

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 $AK = BF$.

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 $H_1H = HH_0 = H_0H_2$.

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 $CDE$.

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

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.

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?

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 $MN // AB$.

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 $XY \perp AC$.

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 $ADC$.

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 $ML$.

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 $ABC$.

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 $\angle R'PB =\angle RPA$.

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.

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.

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.

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 $AK//BC$.

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  $\ell$ .

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?

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.

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

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.

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.

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

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.

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 $EK=KO$.

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?

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 $C_1C_2||O_1O_2$.

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.

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.

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.

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 $XYZ$.

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.

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 $B$.

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.

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 $\angle BAK = \angle CAL$.

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 $6A-2B$.

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.

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.

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.

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?

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.

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.

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

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

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.

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.

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 $A_0C_0 = B_0D_0$.

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 $MT = MC$.

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 $Q$.

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

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.

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 $AC$

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

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 $PP_1 = TT_1$.

Morteza marks six points in the plane. He then calculates and writes down the area of every triangle with vertices in these points ($20$ numbers). Is it possible that all of these numbers are integers, and that they add up to $2019$?

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 minimal possible length of the longest of three segments $B_iO$.
Find the maximal possible length of the shortest of three segments $B_iO$.

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.

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 $PKL$.

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 $\omega$.

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.

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.

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 $BC \cdot AD = 4IL \cdot IN$.

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.

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 $PH \parallel BC$.

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

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 $AC$

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 (i.e. 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.

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

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 $AB$.

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.

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)

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 $\angle XOY=\angle AOB$

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

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?

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 $CF=AH$.

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.

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 $OA$.

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.

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 $Q$.

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.)

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 $$\frac {h_1+h_2- 2R}{h_1h_2}=\frac {h_3+h_4-2R}{h_3h_4}$$

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 $270^{\circ}$?

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 $ABC$.

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.

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 $A'K$.

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 $XZ = YT$.

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.

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.)

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.

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 $FDA$.

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 $G$.

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?

sources:   geometry.ru/olimp/olimpsharygin.php, sasja.shap.homedns.org/Turniry/Shar-eng/