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Moscow Oral Geo 2003-22 228p

geometry problems from Oral Moscow* City Geometry Olympiad
with aops links in the names

mistakenly it was called by me Oral Sharygin Geometry Olympiad,
but it is a Qualification / Regional Round that happens in Moscow
for the Sharygin Geometry Olympiad


juniors collected inside aops here

it didn't take place in 2020

2003 - 2019, 2022

Construct a triangle given an angle, the side opposite the angle and the median to the other side (researching the number of solutions is not required).

In a convex quadrilateral $ABCD$, $\angle ABC = 90^o$ , $\angle BAC = \angle  CAD$, $AC = AD, DH$ is the alltitude of the triangle $ACD$. In what ratio does the line $BH$ divide the segment $CD$?


Inside the segment $AC$, an arbitrary point $B$ is selected and circles with diameters $AB$ and $BC$ are constructed. Points $M$ and $L$ are chosen on the circles (in one half-plane with respect to $AC$), respectively, so that $\angle MBA =  \angle LBC$. Points $K$ and $F$ are marked, respectively, on rays $BM$ and $BL$ so that $BK = BC$ and $BF = AB$. Prove that points $M, K, F$ and $L$ lie on the same circle.

In triangle $ABC$, $M$ is the point of intersection of the medians, $O$ is the center of the inscribed circle, $A', B', C'$ are the points of its tangency with the sides $BC, CA, AB$, respectively. Prove that if $CA'= AB$, then $OM$ and $AB$ are perpendicular.

Given triangle $ABC$. Point $O_1$ is the center of the $BCDE$ rectangle, constructed so that the side $DE$ of the rectangle contains the vertex $A$ of the triangle. Points $O_2$ and $O_3$ are the centers of rectangles constructed in the same way on the sides $AC$ and $AB$, respectively. Prove that lines $AO_1, BO_2$ and $CO_3$ meet at one point.

A circle is located on the plane. What is the smallest number of lines you need to draw so that, symmetrically reflecting a given circle relative to these lines (in any order a finite number of times), it could cover any given point of the plane?

original wording
На плоскости расположен круг. Какое наименьшее количество прямых надо провести, чтобы, симметрично отражая данный круг относительно этих прямых (в любом порядке конечное количество раз), можно было накрыть им любую заданную точку плоскости?


In a convex quadrilateral $ABCD$, $E$ is the midpoint of $CD$, $F$ is midpoint of $AD$, $K$ is the intersection point of $AC$ with $BE$. Prove that the area of triangle $BKF$ is half the area of triangle ABC$.

Construct a triangle $ABC$ given angle $A$ and the medians drawn from vertices $B$ and $C$.

Given a square $ABCD$. Find the locus of points $M$ such that $\angle AMB = \angle CMD$.

Triangle $ABC$ is inscribed in a circle. Through points $A$ and $B$ tangents to this circle are drawn, which intersect at point $P$. Points $X$ and $Y$ are orthogonal projections of point $P$ onto lines $AC$ and $BC$. Prove that line $XY$ is perpendicular to the median of triangle $ABC$ from vertex $C$.

The diagonals of the inscribed quadrilateral $ABCD$ meet at the point $M$, $\angle AMB = 60^o$. Equilateral triangles $ADK$ and $BCL$ are built outward on sides $AD$ and $BC$. Line $KL$ meets the circle circumscribed ariound $ABCD$ at points $P$ and $Q$. Prove that $PK = LQ$.

The length of each side and each non-principal diagonal of a convex hexagon does not exceed $1$. Prove that this hexagon contains a principal diagonal whose length does not exceed $\frac{2}{\sqrt3}$.

$E$ and $F$ are the midpoints of the sides $BC$ and $AD$ of the convex quadrilateral $ABCD$. Prove that the segment $EF$ divides the diagonals $AC$ and $BD$ in the same ratio.

Is there a closed self-intersecting broken line in space that intersects each of its links exactly once, and in it middle of it?

On the board was drawn a circle with a marked center, a quadrangle inscribed in it, and a circle inscribed in it, also with a marked center. Then they erased the quadrilateral (keeping one vertex) and the inscribed circle (keeping its center). Restore any of the erased vertices of the quadrilateral using only a ruler and no more than six lines.

In triangle $ABC$, $M$ is the point of intersection of the medians, $O$ is the center of the inscribed circle. Prove that if the line $OM$ is parallel to the side $BC$, then the point $O$ is equidistant from the sides $AB$ and $AC$.

Trapezoid $ABCD$ with bases $AB$ and $CD$ is inscribed in a circle. Prove that the quadrilateral formed by orthogonal projections of any point of this circle onto lines $AC, BC, AD$ and $BD$ is inscribed.

In the tetrahedron $DABC$ : $\angle ACB = \angle ADB$, $(CD) \perp (ABC)$. In triangle $ABC$, the altitude $h$ drawn to the side $AB$ and the distance $d$ from the center of the circumscribed circle to this side are given. Find the length of the $CD$.



The hexagon has five $90^o$ angles and one $270^o$ angle (see picture). Use a straight-line ruler to divide it into two equal-sized polygons.
A parallelogram of $ABCD$ is given. Line parallel to $AB$ intersects the bisectors of angles $A$ and $C$ at points $P$ and $Q$, respectively. Prove that the angles $ADP$ and $ABQ$ are equal.

(A. Hakobyan)
In triangle $ABC$, points $K ,P$ are chosen on the side $AB$ so that $AK = BL$, and points $M,N$ are chosen on the side $BC$ so that $CN = BM$. Prove that $KN + LM \ge AC$.

(I. Bogdanov)
Given a hexagon $ABCDEF$, in which $AB = BC, CD = DE, EF = FA$, and angles $A$ and $C$ are right. Prove that lines $FD$ and $BE$ are perpendicular.

(B. Kukushkin)
The triangle $ABC$ is inscribed in the circle. Construct a point $P$ such that the points of intersection of lines $AP, BP$ and $CP$ with this circle are the vertices of an equilateral triangle.

(A. Zaslavsky)
Let $A_1,B_1,C_1$ are the midpoints of the sides of the triangle $ABC, I$ is the center of the circle inscribed in it. Let $C_2$ be the point of intersection of lines $C_1 I$ and $A_1B_1$. Let $C_3$ be the point of intersection of lines $CC_2$ and $AB$. Prove that line $IC_3$ is perpendicular to line $AB$.

(A. Zaslavsky)
Given an acute-angled triangle $ABC$. A straight line parallel to $BC$ intersects sides $AB$ and $AC$ at points $M$ and $P$, respectively. At what location of the points $M$ and $P$ will the radius of the circle circumscribed about the triangle $BMP$ be the smallest?
(I. Sharygin)
On a circle with diameter $AB$, lie points $C$ and $D$. $XY$ is the diameter passing through the midpoint $K$ of the chord $CD$. Point $M$ is the projection of point $X$ onto line $AC$, and point $N$ is the projection of point $Y$ on line $BD$. Prove that points $M, N$ and $K$ are collinear.

(A. Zaslavsky)
$ABCBE$ is a regular pentagon. Point $B'$ is symmetric to point $B$ wrt line $AC$ (see figure). Is it possible to pave the plane with pentagons equal to $AB'CBE$?
(S. Markelov)
 
A sphere can be inscribed into a pyramid, the base of which is a parallelogram. Prove that the sums of the areas of its opposite side faces are equal.
(M. Volchkevich)
An arbitrary point M is chosen inside the triangle ABC. Prove that $MA + MB + MC \le max (AB + BC, BC + AC, AC + AB)$.
(N. Sedrakyan)
Six straight lines are drawn on the plane. It is known that for any three of them there is a fourth of the same set of lines, such that all four will touch some circle. Do all six lines necessarily touch the same circle?
(I. Bogdanov)


The diagonals of the inscribed quadrangle $ABCD$ intersect at point $K$. Prove that the tangent at point $K$ to the circle circumscribed around the triangle $ABK$ is parallel to $CD$.

(A Zaslavsky)
Determine the ratio of the sides of the rectangle circumscribed around a corner of five cells (see figure).

(M. Evdokimov)
On the sides $AB, BC$ and $AC$ of the triangle $ABC$, points $C', A'$ and $B'$ are selected, respectively, so that the angle $A'C'B'$ is right. Prove that the segment $A'B'$ is longer than the diameter of the inscribed circle of the triangle $ABC$.
(M. Volchkevich)
An arbitrary triangle $ABC$ is given. Construct a straight line passing through vertex $B$ and dividing it into two triangles, the radii of the inscribed circles of which are equal.
(M. Volchkevich)
Equilateral triangles $ABC_1, BCA_1, CAB_1$ are built on the sides of the triangle $ABC$ to the outside. On the segment $A_1B_1$ to the outer side of the triangle $A_1B_1C_1$, an equilateral triangle $A_1B_1C_2$ is constructed. Prove that $C$ is the midpoint of the segment $C_1C_2$.

(A. Zaslavsky)
In an acute-angled triangle, one of the angles is $60^o$. Prove that the line passing through the center of the circumcircle and the point of intersection of the medians of the triangle cuts off an equilateral triangle from it.

(A. Zaslavsky)
An arbitrary triangle $ABC$ is given. Construct a line that divides it into two polygons, which have equal radii of the circumscribed circles.
(L. Blinkov)
Six segments are such that any three can form a triangle. Is it true that these segments can be used to form a tetrahedron?
(S. Markelov)
Two non-rolling circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$and radii $2R$ and $R$, respectively, are given on the plane. Find the locus of the centers of gravity of triangles in which one vertex lies on $C_1$ and the other two lie on $C_2$.

(B. Frenkin)
The quadrangle $ABCD$ is inscribed in a circle, the center $O$ of which lies inside it. The tangents to the circle at points $A$ and $C$ and a straight line, symmetric to $BD$ wrt point $O$, intersect at one point. Prove that the products of the distances from $O$ to opposite sides of the quadrilateral are equal.

(A. Zaslavsky)
The base of the pyramid is a convex quadrangle. Is there necessarily a section of this pyramid that does not intersect the base and is an inscribed quadrangle?
(M. Volchkevich)
Given triangle $ABC$ and points $P$. Let $A_1,B_1,C_1$ be the second points of intersection of straight lines $AP, BP, CP$ with the circumscribed circle of $ABC$. Let points $A_2, B_2, C_2$ be symmetric to $A_1,B_1,C_1$ wrt $BC,CA,AB$, respectively. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are similar.
(A. Zaslavsky)

Given a rectangular strip of measure $12 \times 1$. Paste this strip in two layers over the cube with edge $1$ (the strip can be bent, but cannot be cut).

(V. Shevyakov)
An isosceles right-angled triangle $ABC$ is given. On the extensions of sides $AB$ and $AC$, behind vertices $B$ and $C$ equal segments $BK$ and $CL$ were laid. $E$ and F are the points of intersection of the segment $KL$ and the lines perpendicular to the $KC$ , passing through the points $B$ and $A$, respectively. Prove that $EF = FL$.

Construct a parallelogram $ABCD$, if three points are marked on the plane: the midpoints of its altitudes $BH$ and $BP$ and the midpoint of the side $AD$.

Let $I$ be the center of a circle inscribed in triangle $ABC$. The circle circumscribed about the triangle $BIC$ intersects lines $AB$ and $AC$ at points $E$ and $F$, respectively. Prove that the line $EF$ touches the circle inscribed in the triangle $ABC$.

Given triangle $ABC$. Points $A_1,B_1$ and $C_1$ are symmetric to its vertices with respect to opposite sides. $C_2$ is the intersection point of lines $AB_1$ and $BA_1$. Points$ A_2$ and $B_2$ are defined similarly. Prove that the lines $A_1 A_2, B_1 B_2$ and $C_1 C_2$ are parallel.

(A. Zaslavsky)
A point $P$ is fixed inside the circle. $C$ is an arbitrary point of the circle, $AB$ is a chord passing through point $B$ and perpendicular to the segment $BC$. Points $X$ and $Y$ are projections of point $B$ onto lines $AC$ and $BC$. Prove that all line segments $XY$ are tangent to the same circle.

(A. Zaslavsky)
The triangle was divided into five triangles similar to it. Is it true that the original triangle is right-angled?

(S. Markelov)
Two circles intersect at points $P$ and $Q$. Point $A$ lies on the first circle, but outside the second. Lines $AP$ and $AQ$ intersect the second circle at points $B$ and $C$, respectively. Indicate the position of point $A$ at which triangle $ABC$ has the largest area.

(D. Prokopenko)
In a trapezoid, the sum of the lengths of the side and the diagonal is equal to the sum of the lengths of the other side and the other diagonal. Prove that the trapezoid is isosceles.

The midpoints of the opposite sides of the hexagon are connected by segments. It turned out that the points of pairwise intersection of these segments form an equilateral triangle. Prove that the drawn segments are equal.
(M. Volchkevich)
At the base of the quadrangular pyramid $SABCD$ lies the quadrangle $ABCD$. whose diagonals are perpendicular and intersect at point $P$, and $SP$ is the altitude of the pyramid. Prove that the projections of the point $P$ onto the lateral faces of the pyramid lie on the same circle.

(A. Zaslavsky)
A circle and a point $P$ inside it are given. Two arbitrary perpendicular rays starting at point $P$ intersect the circle at points $A$ and $B$. Point $X$ is the projection of point $P$ onto line $AB, Y$ is the point of intersection of tangents to the circle drawn through points $A$ and $B$. Prove that all lines $XY$ pass through the same point.
(A. Zaslavsky)

A coordinate system was drawn on the board and points $A (1,2)$ and $B (3,1)$ were marked. The coordinate system was erased. Restore it by the two marked points.

In a certain triangle, the bisectors of the two interior angles were extended to the intersection with the circumscribed circle and two equal chords were obtained. Is it true that the triangle is isosceles?

In the regular hexagon $ABCDEF$ on the line $AF$, the point $X$ is taken so that the angle $XCD$ is $45^o$. Find the angle $FXE$.

(Kiev Olympiad)
A circle can be described around the quadrilateral $ABCD$. Point $P$ is the foot of the perpendicular dropped from point $A$ on line $BC$, and respectively $Q$ from $A$ on $DC$, $R$ from $D$ on $AB$ and $T$ from $D$ on $BC$ . Prove that points $P,Q,R$ and $T$ lie on the same circle.

(A. Myakishev)
Reconstruct an acute-angled triangle given the orthocenter and midpoints of two sides.

(A. Zaslavsky)
Opposite sides of a convex hexagon are parallel. Let's call the "height" of such a hexagon a segment with ends on straight lines containing opposite sides and perpendicular to them. Prove that a circle can be circumscribed around this hexagon if and only if its "heights" can be parallelly moved so that they form a triangle.

(A. Zaslavsky)
Each of two similar triangles was cut into two triangles so that one of the resulting parts of one triangle is similar to one of the parts of the other triangle. Is it true that the remaining parts are also similar?

(D. Shnol)
The radii $r$ and $R$ of two non-intersecting circles are given. The common internal tangents of these circles are perpendicular. Find the area of the triangle bounded by these tangents, as well as the common external tangent.

Given a quadrilateral $ABCD$. $A ', B', C'$ and $D'$ are the midpoints of the sides $BC, CB, BA$ and $AB$, respectively. It is known that $AA'= CC', BB'= DD'$. Is it true that $ABCD$ is a parallelogram?

(M. Volchkevich)
Angle $A$ in triangle $ABC$ is equal to $120^o$. Prove that the distance from the center of the circumscribed circle to the orthocenter is equal to $AB + AC$.
(V. Protasov)
There are two shawls, one in the shape of a square, the other in the shape of a regular triangle, and their perimeters are the same. Is there a polyhedron that can be completely pasted over with these two shawls without overlap (shawls can be bent, but not cut)?
(S. Markelov)
Given a triangle $ABC$ and points $P$ and $Q$. It is known that the triangles formed by the projections $P$ and $Q$ on the sides of $ABC$ are similar (vertices lying on the same sides of the original triangle correspond to each other). Prove that line $PQ$ passes through the center of the circumscribed circle of triangle $ABC$.
(A. Zaslavsky)


The figure shows a parallelogram and the point $P$ of intersection of its diagonals is marked. Draw a straight line through $P$ so that it breaks the parallelogram into two parts, from which you can fold a rhombus.
A square and a rectangle of the same perimeter have a common corner. Prove that the intersection point of the diagonals of the rectangle lies on the diagonal of the square.
(Yu. Blinkov)
In the triangle $ABC$, $AA_1$ and $BB_1$ are altitudes. On the side $AB$ , points $M$ and $K$ are selected so that $B_1K \parallel BC$ and $A_1M \parallel AC$. Prove that the angle $AA_1K$ is equal to the angle $BB_1M$.
(D. Prokopenko)
Construct a triangle given a side, the radius of the inscribed circle, and the radius of the exscribed circle tangent to that side. (Research is not required.)

A treasure is buried at some point on a round island with a radius of $1$ km. On the coast of the island there is a mathematician with a device that indicates the direction to the treasure when the distance to the treasure does not exceed $500$ m. In addition, the mathematician has a map of the island, on which he can record all his movements, perform measurements and geometric constructions. The mathematician claims that he has an algorithm for how to get to the treasure after walking less than $4$ km. Could this be true?
(B. Frenkin)
Fixed two circles $w_1$ and $w_2$, $\ell$ one of their external tangent and $m$ one of their internal tangent . On the line $m$, a point $X$ is chosen, and on the line $\ell$, points $Y$ and $Z$ are constructed so that $XY$ and $XZ$ touch $w_1$ and $w_2$, respectively, and the triangle $XYZ$ contains circles $w_1$ and $w_2$. Prove that the centers of the circles inscribed in triangles $XYZ$ lie on one line.
(P. Kozhevnikov)
2009 Oral Moscow Geometry Olympiad grades 10-11 p1
Are there two such quadrangles that the sides of the first are less than the corresponding sides of the second, and the corresponding diagonals are larger?
(Arseniy Akopyan)
2009 Oral Moscow Geometry Olympiad grades 10-11 p2
Trapezium $ABCD$ and parallelogram $MBDK$ are located so that the sides of the parallelogram are parallel to the diagonals of the trapezoid (see fig.). Prove that the area of the gray part is equal to the sum of the areas of the black part.
(Yu. Blinkov)
Altitudes $AA_1$ and $BB_1$ are drawn in the acute-angled triangle $ABC$. Prove that the perpendicular dropped from the point of tangency of the inscribed circle with the side $BC$, on the line $AC$ passes through the center of the inscribed circle of the triangle $A_1CB_1$.

(V. Protasov)
Three circles are constructed on the medians of a triangle as on diameters. It is known that they intersect in pairs. Let $C_1$ be the point of intersection of the circles built on the medians $AM_1$ and $BM_2$, which is more distant from the vertex $C$. Points $A_1$ and $B_1$ are defined similarly. Prove that they lines $AA_1, BB_1$ and $CC_1$ intersect at one point.
(D. Tereshin)
Prove that any convex polyhedron has three edges that can be used to form a triangle.

(Barbu Bercanu, Romania)
To two circles $r_1$ and $r_2$, intersecting at points $A$ and $B$, their common tangent $CD$ is drawn ($C$ and $D$ are tangency points, respectively, point $B$ is closer to line $CB$ than $A$). Line passing through $A$ , intersects $r_1$ and $r_2$ for second time at points $K$ and $L$, respectively ($A$ lies between $K$ and $L$). Lines $KC$ and $LD$ intersect at point $P$. Prove that $PB$ is the symmetian of triangle $KPL$.
(Yu. Blinkov)
Two equilateral triangles $ABC$ and $CDE$ have a common vertex (see fig). Find the angle between straight lines $AD$ and $BE$.
Given a square sheet of paper with side $1$. Measure on this sheet a distance of $ 5/6$.
(the sheet can be folded, including, along any segment with ends at the edges of the paper and unbend back, after unfolding, a trace of the fold line remains on the paper).

Two circles $w_1$ and $w_2$ intersect at points $A$ and $B$. Tangents $\ell_1$ and $\ell_2$ respectively are drawn to them through point $A$. The perpendiculars dropped from point $B$ to $\ell_2$ and $\ell_1$ intersects the circles $w_1$ and $w_2$, respectively, at points $K$ and $N$. Prove that points $K, A$ and $N$ lie on one straight line.

An isosceles triangle $ABC$ with base $AC$ is given. Point $H$ is the intersection of altitudes. On the sides $AB$ and $BC$, points $M$ and $K$ are selected, respectively, so that the angle $KMH$ is right. Prove that a right-angled triangle can be constructed from the segments $AK, CM$ and $MK$.

Points $K$ and $M$ are taken on the sides $AB$ and $CD$ of square $ABCD$, respectively, and on the diagonal $AC$ - point $L$ such that $ML = KL$. Let $P$ be the intersection point of the segments $MK$ and $BD$. Find the angle $KPL$.

Perpendicular bisectors of the sides $BC$ and $AC$ of an acute-angled triangle $ABC$ intersect lines $AC$ and $BC$ at points $M$ and $N$. Let point $C$ move along the circumscribed circle of triangle $ABC$, remaining in the same half-plane relative to $AB$ (while points $A$ and $B$ are fixed). Prove that line $MN$ touches a fixed circle.

2010 Oral Moscow Geometry Olympiad grades 10-11 p1
Convex $n$-gon $P$, where $n> 3$, is cut into equal triangles by diagonals that do not intersect inside it. What are the possible values of $n$ if the $n$-gon is cyclic?
Quadrangle $ABCD$ is inscribed in a circle. The perpendicular from the vertex $C$ on the bisector of $\angle ABD$ intersects the line $AB$ at the point $C_1$. The perpendicular from the vertex $B$ on the bisector of $\angle ACD$ intersects the line $CD$ at the point $B_1$. Prove that $B_1C_1 \parallel AD$.

On the sides $AB$ and $BC$ of triangle $ABC$, points $M$ and $K$ are taken, respectively, so that $S_{KMC} + S_{KAC}=S_{ABC}$. Prove that all such lines $MK$ pass through one point.

From the vertice $A$ of the parallelogram $ABCD$, the perpendiculars $AM,AN$ on sides $BC,CD$ respectively. $P$ is the intersection point of $BN$ and $DM$. Prove that the lines $AP$ and $MN$ are perpendicular.

All edges of a regular right pyramid are equal to $1$, and all vertices lie on the side surface of a (infinite) right circular cylinder of radius $R$. Find all possible values of $R$.

In a triangle $ABC, O$ is the center of the circumscribed circle. Line $a$ passes through the midpoint of the altitude of the triangle from the vertex $A$ and is parallel to $OA$. Similarly, the straight lines $b$ and $c$ are defined. Prove that these three lines intersect at one point.


2011 Oral Moscow Geometry Olympiad grades 8-9 p1
The bisector of angle $B$ and the bisector of external angle $D$ of rectangle $ABCD$ intersect side $AD$ and line $AB$ at points $M$ and $K$, respectively. Prove that the segment $MK$ is equal and perpendicular to the diagonal of the rectangle.

2011 Oral Moscow Geometry Olympiad grades 8-9 p2
In an isosceles triangle $ABC$ ($AB=AC$) on the side $BC$, point $M$ is marked so that the segment $CM$ is equal to the altitude of the triangle drawn on this side, and on the side $AB$, point $K$ is marked so that the angle $\angle KMC$ is right. Find the angle $\angle ACK$.

2011 Oral Moscow Geometry Olympiad grades 8-9 p3
A $2\times 2$ square was cut from a squared sheet of paper. Using only a ruler without divisions and without going beyond the square, divide the diagonal of the square into $6$ equal parts.

2011 Oral Moscow Geometry Olympiad grades 8-9 p4
In the trapezoid $ABCD,  AB = BC = CD, CH$ is the altitude. Prove that the perpendicular from $H$ on $AC$ passes through the midpoint of $BD$.

2011 Oral Moscow Geometry Olympiad grades 8-9 p5
Let $AA _1$ and $BB_1$ be the altitudes of an isosceles acute-angled triangle $ABC, M$ the middle of $AB$. The circles circumscribed around the triangles $AMA_1$ and $BMB_1$ intersect the lines $AC$ and $BC$ at points $K$ and $L$, respectively. Prove that $K, M$, and $L$ lie on the same line.

2011 Oral Moscow Geometry Olympiad grades 8-9 p6
One triangle lies inside another. Prove that at least one of the two smallest sides (out of six) is the side of the inner triangle.

2011 Oral Moscow Geometry Olympiad grades 10-11 p1
$AD$ and $BE$ are the altitudes of the triangle $ABC$. It turned out that the point $C'$, symmetric to the vertex $C$ wrt to the midpoint of the segment $DE$, lies on the side $AB$. Prove that $AB$ is tangent to the circle circumscribed around the triangle $DEC'$.

2011 Oral Moscow Geometry Olympiad grades 10-11 p2
Line $\ell $ intersects the plane $a$. It is known that in this plane there are $2011$ straight lines equidistant from $\ell$ and not intersecting $\ell$. Is it true that $\ell$ is perpendicular to $a$?

2011 Oral Moscow Geometry Olympiad grades 10-11 p3
A non-isosceles trapezoid $ABCD$ ($AB // CD$) is given. An arbitrary circle passing through points $A$ and $B$ intersects the sides of the trapezoid at points $P$ and $Q$, and the intersect the diagonals at points $M$ and $N$. Prove that the lines $PQ, MN$ and $CD$ are concurrent.

2011 Oral Moscow Geometry Olympiad grades 10-11 p4
Prove that any rigid flat triangle $T$ of area less than $4$ can be inserted through a triangular hole $Q$ with area $3$.

2011 Oral Moscow Geometry Olympiad grades 10-11 p5
In a convex quadrilateral $ABCD, AC\perp BD, \angle BCA = 10^o,\angle BDA = 20^o, \angle BAC = 40^o$. Find $\angle BDC$.

2011 Oral Moscow Geometry Olympiad grades 10-11 p6
Let $AA_1 , BB_1$, and $CC_1$ be the altitudes of the non-isosceles acute-angled triangle $ABC$. The circles circumscibred around the triangles $ABC$ and $A_1 B_1 C$ intersect again at the point $P , Z$ is the intersection point of the tangents to the circumscribed circle of the triangle $ABC$ conducted at points $A$ and $B$ . Prove that lines $AP , BC$ and $ZC_1$ are concurrent.


2012 Oral Moscow Geometry Olympiad grades 8-9 p1
In trapezoid $ABCD$, the sides $AD$ and $BC$ are parallel, and $AB = BC = BD$. The height $BK$ intersects the diagonal $AC$ at $M$. Find $\angle CDM$.

2012 Oral Moscow Geometry Olympiad grades 8-9 p2 
Two equal polygons $F$ and $F'$ are given on the plane. It is known that the vertices of the polygon $F$ belong to $F'$ (may lie inside it or on the border). Is it true that all the vertices of these polygons coincide?

2012 Oral Moscow Geometry Olympiad grades 8-9 p3
Given an equilateral triangle $ABC$ and a straight line $\ell$, passing through its center. Intersection points of this line with sides $AB$ and $BC$ are reflected wrt to the midpoints of these sides respectively. Prove that the line passing through the resulting points, touches the inscribed circle triangle $ABC$.

2012 Oral Moscow Geometry Olympiad grades 8-9 p4
In triangle $ABC$, point $I$ is the center of the inscribed circle points, points $I_A$ and $I_C$ are the centers of the excircles, tangent to sides $BC$ and $AB$, respectively. Point $O$ is the center of the circumscribed circle of triangle $II_AI_C$. Prove that $OI \perp AC$

2012 Oral Moscow Geometry Olympiad grades 8-9 p5
Given a circle and a chord $AB$, different from the diameter. Point $C$ moves along the large arc $AB$. The circle passing through passing through points $A, C$ and point $H$ of intersection of heights of of the triangle $ABC$, re-intersects the line $BC$ at point $P$. Prove that line $PH$ passes through a fixed point independent of the position of point $C$.

Restore the triangle with a compass and a ruler given the point of intersection of heights and the bases of the median and bisectors drawn to one side. (No research required.)

2012 Oral Moscow Geometry Olympiad grades 10-11 p1
Is it true that the center of the inscribed circle of the triangle lies inside the triangle formed by the lines of connecting it's midpoints?

2012 Oral Moscow Geometry Olympiad grades 10-11 p2
In the convex pentagon $ABCDE$: $\angle A = \angle C = 90^o$, $AB = AE, BC = CD, AC = 1$. Find the area of the pentagon.

2012 Oral Moscow Geometry Olympiad grades 10-11 p3
$H$ is the intersection point of the heights $AA'$ and $BB'$ of the acute-angled triangle $ABC$. A straight line, perpendicular to $AB$, intersects these heights at points $D$ and $E$, and side $AB$ at point $P$. Prove that the orthocenter of the triangle $DEH$ lies on segment $CP$.
2012 Oral Moscow Geometry Olympiad grades 10-11 p4
Inside the convex polyhedron, the point $P$ and several lines $\ell_1,\ell_2, ..., \ell_n$ passing through $P$ and not lying in the same plane. To each face of the polyhedron we associate one of the lines $l_1, l_2, ..., l_n$ that forms the largest angle with the plane of this face (if there are there are several direct ones, we will choose any of them). Prove that there is a face that intersects with its corresponding line.
2012 Oral Moscow Geometry Olympiad grades 10-11 p5
Inside the circle with center $O$, points $A$ and $B$ are marked so that $OA = OB$. Draw a point $M$ on the circle from which the sum of the distances to points $A$ and $B$ is the smallest among all possible.
Tangents drawn to the circumscribed circle of an acute-angled triangle $ABC$ at points $A$ and $C$, intersect at point $Z$. Let $AA_1, CC_1$ be heights. Line $A_1C_1$ intersects $ZA, ZC$ at points $X$ and $Y$, respectively. Prove that the circumscribed circles of the triangles $ABC$ and $XYZ$ are tangent.



2013 Oral Moscow Geometry Olympiad grades 8-9 p1
In triangle $ABC$ the bisector $AK$ is perpendicular on the median is $CL$. Prove that in the triangle $BKL$ also one of bisectors are perpendicular to one of the medians.

2013 Oral Moscow Geometry Olympiad grades 8-9 p2
With a compass and a ruler, split a triangle into two smaller triangles with the same sum of squares of sides.

2013 Oral Moscow Geometry Olympiad grades 8-9 p3
The bisectors $AA_1$ and $CC_1$ of the right triangle $ABC$ ($\angle B = 90^o$) intersect at point $I$. The line passing through the point $C_1$ and perpendicular on the line $AA_1$ intersects the line that passes through $A_1$ and is perpendicular on $CC_1$, at the point $K$. Prove that the midpoint of the segment $KI$ lies on segment $AC$.

2013 Oral Moscow Geometry Olympiad grades 8-9 p4
Let $ABC$ be a triangle. On the extensions of sides $AB$ and $CB$ towards $B$, points $C_1$ and $A_1$ are taken, respectively, so that $AC = A_1C = AC_1$. Prove that circumscribed circles of triangles $ABA_1$ and $CBC_1$ intersect on the bisector of angle $B$.

2013 Oral Moscow Geometry Olympiad grades 8-9 p5
In triangle $ABC, \angle C= 60^o, \angle A= 45^o$. Let $M$ be the midpoint of $BC, H$ be the orthocenter of triangle $ABC$. Prove that line $MH$ passes through the midpoint of arc $AB$ of the circumcircle of triangle $ABC$.

Let $ABC$ be a triangle. On its sides $AB$ and $BC$ are fixed points $C_1$ and $A_1$, respectively. Find a point$ P$ on the circumscribed circle of triangle $ABC$ such that the distance between the centers of the circumscribed circles of the triangles $APC_1$ and $CPA_1$ is minimal.

2013 Oral Moscow Geometry Olympiad grades 10-11 p1
Diagonals of an cyclic quadrilateral $ABCD$ intersect at point $O$. The circumscribed circles of triangles $AOB$ and $COD$ intersect at point $M$ on the $AD$ side. Prove that the point $O$ is the center of the inscribed circle of the triangle $BMC$.

2013 Oral Moscow Geometry Olympiad grades 10-11 p2
Inside the angle $AOD$, the rays $OB$ and $OC$ are drawn  such that $\angle AOB = \angle COD.$ Two circles are inscribed inside the angles $\angle AOB$ and $\angle COD$ . Prove that the intersection point of the common internal tangents of these circles lies on the bisector of the angle $AOD$.

2013 Oral Moscow Geometry Olympiad grades 10-11 p3
Is there a polyhedron whose area ratio of any two faces is at least $2$ ?

2013 Oral Moscow Geometry Olympiad grades 10-11 p4
Similar triangles $ABM, CBP, CDL$ and $ADK$ are built on the sides of the quadrilateral $ABCD$ with perpendicular diagonals in the outer side (the neighboring ones are oriented differently). Prove that $PK = ML$.

2013 Oral Moscow Geometry Olympiad grades 10-11 p5
In the acute-angled triangle $ABC$, let  $AP$ and $BQ$ be the heights,  $CM$ be the median . Point $R$ is the midpoint of $CM$. Line $PQ$ intersects line $AB$ at $T$. Prove that $OR \perp TC$, where $O$ is the center of the circumscribed circle of triangle $ABC$.

The trapezoid $ABCD$ is inscribed in the circle $w$ ($AD // BC$). The circles inscribed in the triangles $ABC$ and $ABD$ touch the base of the trapezoid $BC$ and  $AD$ at points $P$ and $Q$ respectively. Points $X$ and $Y$ are the midpoints of the arcs $BC$ and $AD$ of circle $w$ that do not contain points $A$ and $B$ respectively. Prove that lines $XP$ and $YQ$ intersect on the circle $w$.



2014 Oral Moscow Geometry Olympiad grades 8-9 p1
In triangle $ABC, \angle A= 45^o, BH$ is the height, the point $K$ lies on the $AC$ side, and $BC = CK$. Prove that the center of the circumscribed circle of triangle $ABK$ coincides with the center of an excircle of triangle $BCH$.

2014 Oral Moscow Geometry Olympiad grades 8-9 p2
Let $ABCD$ be a parallelogram. On side $AB$, point $M$ is taken so that $AD = DM$.
On side $AD$ point $N$ is taken so that $AB = BN$. Prove that $CM = CN$.

2014 Oral Moscow Geometry Olympiad grades 8-9 p3
Is there a convex pentagon in which each diagonal is equal to a side?

2014 Oral Moscow Geometry Olympiad grades 8-9 p4
In triangle $ABC$, the perpendicular bisectors of sides $AB$ and $BC$ intersect side $AC$ at points $P$ and $Q$, respectively, with point $P$ lying on the segment $AQ$. Prove that the circumscribed circles of the triangles $PBC$ and $QBA$ intersect on the bisector of the angle $PBQ$.

2014 Oral Moscow Geometry Olympiad grades 8-9 p5
Segment $AD$ is the diameter of the circumscribed circle of an acute-angled triangle $ABC$. Through the intersection of the heights of this triangle, a straight line was drawn parallel to the side $BC$, which intersects sides $AB$ and $AC$ at points $E$ and $F$, respectively. Prove that the perimeter of the triangle $DEF$ is two times larger than the side $BC$.

Inside an isosceles right triangle $ABC$ with hypotenuse $AB$ a point $M$ is taken such that the angle $MAB$ is $15 ^o$ larger than the $MAC$ angle, and the $MCB$ angle is $15^o$ larger than the angle $MBC$. Find the $BMC$ angle.

2014 Oral Moscow Geometry Olympiad grades 10-11 p1
In trapezoid $ABCD$: $BC <AD, AB = CD, K$ is midpoint of $AD, M$ is midpoint of $CD, CH$ is height. Prove that lines $AM, CK$ and $BH$ intersect at one point.

2014 Oral Moscow Geometry Olympiad grades 10-11 p2
Is it possible to cut a regular triangular prism into two equal pyramids?

2014 Oral Moscow Geometry Olympiad grades 10-11 p3
The bisectors $AA_1$ and $CC_1$ of triangle $ABC$ intersect at point $I$. The circumscribed circles of triangles $AIC_1$ and $CIA_1$ intersect the arcs $AC$ and $BC$ (not containing points $B$ and $A$ respectively) of the circumscribed circle of triangle $ABC$ at points $C_2$ and $A_2$, respectively. Prove  that lines $A_1A_2$ and $C_1C_2$ intersect on the circumscribed circle of triangle $ABC$.

2014 Oral Moscow Geometry Olympiad grades 10-11 p4
The medians $AA_0, BB_0$, and $CC_0$ of the acute-angled triangle $ABC$ intersect at the point $M$, and heights $AA_1, BB_1$ and $CC_1$  at point $H$. Tangent to the circumscribed circle of triangle $A_1B_1C_1$ at $C_1$ intersects the line $A_0B_0$ at the point C'. Points $A'$  and $B'$ are defined similarly. Prove that $A', B'$  and $C'$ lie on one line perpendicular to the line $MH$.

2014 Oral Moscow Geometry Olympiad grades 10-11 p5  (also USAJMO 2017)
Given a regular triangle $ABC$, whose area is $1$, and the point $P$ on its circumscribed circle. Lines $AP, BP, CP$ intersect, respectively, lines $BC, CA, AB$ at points $A', B', C'$. Find the area of the triangle $A'B'C'$.

A convex quadrangle $ABCD$ is given. Let $I$ and $J$ be the circles of circles inscribed in the triangles $ABC$ and $ADC$, respectively, and $I_a$ and $J_a$ are the centers of the excircles circles of triangles $ABC$ and $ADC$, respectively (inscribed in the angles $BAC$ and $DAC$, respectively). Prove that the intersection point $K$ of the lines $IJ_a$ and $JI_a$ lies on the bisector of the angle $BCD$.



2015 Oral Moscow Geometry Olympiad grades 8-9 p1
In triangle $ABC$, the height $AH$ passes through middle of the median $BM$. Prove that in the triangle $BMC$ also one of the heights passes through the middle of one of the medians.

2015 Oral Moscow Geometry Olympiad grades 8-9 p2
The square $ABCD$ and the equilateral triangle $MKL$ are located as shown in the figure. Find the angle $PQD$.

2015 Oral Moscow Geometry Olympiad grades 8-9 p3
In triangle $ABC$, points $D, E$, and $F$ are marked on sides $AC, BC$, and $AB$ respectively, so that $AD = AB, EC = DC, BF = BE$. After that, they erased everything except points $E, F$ and $D$. Reconstruct the $ABC$ triangle (no study required).

2015 Oral Moscow Geometry Olympiad grades 8-9 p4
In trapezoid $ABCD$, the bisectors of angles $A$ and $D$ intersect at point $E$ lying on the side of $BC$. These bisectors divide the trapezoid into three triangles into which the circles are inscribed. One of these circles touches the base $AB$ at the point $K$, and two others touch the bisector $DE$ at points $M$ and $N$. Prove that $BK = MN$.

2015 Oral Moscow Geometry Olympiad grades 8-9 p5
On the $BE$ side of a regular $ABE$ triangle, a $BCDE$ rhombus is built outside it. The segments $AC$ and $BD$ intersect at point $F$. Prove that $AF <BD$.

In the acute-angled non-isosceles triangle $ABC$, the height $AH$ is drawn. Points $B_1$ and $C_1$ are marked on the sides $AC$ and $AB$, respectively, so that $HA$ is the angle bisector of $B_1HC_1$ and quadrangle $BC_1B_1C$ is cyclic. Prove that $B_1$ and $C_1$ are base of the heights of triangle $ABC$.

2015 Oral Moscow Geometry Olympiad grades 10-11 p1
Two trapezoid angles and diagonals are respectively equal. Is it true that such are the trapezoid equal?

2015 Oral Moscow Geometry Olympiad grades 10-11 p2
Line $\ell$ is perpendicular to one of the medians of the triangle. The perpendicular bisectors of the sides of this triangle intersect line $\ell$ at three points. Prove that one of them is the middle of the segment formed by the remaining two.

2015 Oral Moscow Geometry Olympiad grades 10-11 p3
$O$ is the intersection point of the diagonals of the trapezoid $ABCD$. A line passing through $C$ and a point symmetric to $B$ with respect to $O$, intersects the base $AD$ at the point $K$. Prove that $S_{AOK} = S_{AOB} + S_{DOK}$.

2015 Oral Moscow Geometry Olympiad grades 10-11 p4
In triangle $ABC$, point $M$ is the midpoint of $BC, P$ is the intersection point of the tangents at points $B$ and $C$ of the circumscribed circle, $N$ is the midpoint of the segment $MP$. The segment $AN$ intersects the circumscribed circle at point $Q$. Prove that $\angle PMQ = \angle MAQ$.

2015 Oral Moscow Geometry Olympiad grades 10-11 p5
A triangle $ABC$ and spheres are given in space $S_1$ and $S_2$, each of which passes through points $A, B$ and $C$. For points $M$ spheres $S_1$ not lying in the plane of triangle $ABC$ are drawn lines $MA, MB$ and $MC$, intersecting the sphere $S_2$ for the second time at points $A_1,B_1$ and $C_1$, respectively. Prove that the planes passing through points $A_1, B_1$ and $C_1$, touch a fixed sphere or pass through a fixed point.

In an acute-angled isosceles triangle $ABC$, altitudes $CC_1$ and $BB_1$ intersect the line passing through the the vertex $A$ and parallel to the line $BC$, at points $P$ and $Q$. Let $A_0$ be the midpoint of side $BC$, and $AA_1$ the altitude. Lines $A_0C_1$ and $A_0B_1$ intersect line $PQ$ at points $K$ and $L$. Prove that the circles circumscribed around triangles $PQA_1, KLA_0, A_1B_1C_1$ and a circle with a diameter $AA_1$ intersect at one point.



2016 Oral Moscow Geometry Olympiad grades 8-9 p1
Angles are equal in a hexagon, three main diagonals are equal and the other six diagonals are also equal. Is it true that it has equal sides?

2016 Oral Moscow Geometry Olympiad grades 8-9 p2
In the rectangle there is a broken line, the neighboring links of which are perpendicular and equal to the smaller side of the rectangle (see the figure). Find the ratio of the sides of the rectangle.


2016 Oral Moscow Geometry Olympiad grades 8-9 p3
A circle with center $O$ passes through the ends of the hypotenuse of a right-angled triangle and intersects its legs at points $M$ and $K$. Prove that the distance from point $O$ to line $MK$ is half the hypotenuse.

2016 Oral Moscow Geometry Olympiad grades 8-9 p4
Let $M$ and $N$ be the midpoints of the hypotenuse $AB$ and the leg $BC$ of a right triangles $ABC$ respectively. The excircle of the triangle $ACM$ touches the side $AM$ at point $Q$, and line $AC$ at point $P$. Prove that points $P, Q$ and $N$ lie on one straight line.

2016 Oral Moscow Geometry Olympiad grades 8-9 p5
Points $I_A, I_B, I_C$ are the centers of the excircles of $ABC$ related to sides $BC, AC$ and $AB$ respectively. Perpendicular from $I_A$ to $AC$ intersects the perpendicular from $I_B$ to $B_C$ at point $X_C$. The points $X_A$ and $X_B$. Prove that the lines $I_AX_A, I_BX_B$ and $I_CX_C$ intersect at the same point.

2016 Oral Moscow Geometry Olympiad grades 8-9 p6
Given a square sheet of paper with a side of $2016$. Is it possible to bend its not more than ten times, build a segment of length $1$?

2016 Oral Moscow Geometry Olympiad grades 10-11 p1
The line passing through the center $I$ of the inscribed circle of a triangle $ABC$, perpendicular to $AI$ and intersects sides $AB$ and $AC$ at points $C'$ and $B'$, respectively. In the triangles $BC'I$ and $CB'I$, the heights $C'C_1$ and $B'B_1$ were drawn, respectively. Prove that the middle of the segment $B_1C_1$ lies on a straight line passing through point $I$ and perpendicular to $BC$.

2016 Oral Moscow Geometry Olympiad grades 10-11 p2
A regular heptagon $A_1A_2A_3A_4A_5A_6A_7$ is given. Straight $A_2A_3$ and $A_5A_6$ intersect at point $X$, and straight lines $A_3A_5$ and $A_1A_6$ - at point Y. Prove that lines $A_1A_2$ and $XY$ are parallel.

2016 Oral Moscow Geometry Olympiad grades 10-11 p3
Two squares are arranged as shown in the picture. Prove that the areas of shaded quadrilaterals are equal.
2016 Oral Moscow Geometry Olympiad grades 10-11 p4
In a convex $n$-gonal prism all sides are equal. For what $n$ is this prism right?

2016 Oral Moscow Geometry Olympiad grades 10-11 p5
From point $A$ to circle $\omega$ tangent $AD$ and arbitrary a secant intersecting a circle at points $B$ and $C$ (B lies between points $A$ and $C$). Prove that the circle passing through points $C$ and $D$ and touching the straight line $BD$, passes through a fixed point (other than $D$).

2016 Oral Moscow Geometry Olympiad grades 10-11 p6
Given an acute triangle $ABC$. Let $A'$ be a point symmetric to $A$ with respect to $BC, O_A$ is the center of the circle passing through $A$ and the midpoints of the segments $A'B$ and $A'C. O_B$ and $O_C$ points  are defined similarly. Find the ratio of the radii of the circles circumscribed around the triangles $ABC$ and $O_AO_BO_C$.



2017 Oral Moscow Geometry Olympiad grades 8-9 p1
On side $AB$ of triangle $ABC$ is marked point $K$ such that $AB = CK$. Points $N$ and $M$ are the midpoints of $AK$ and $BC$, respectively. The segments $NM$ and $CK$ intersect in point $P$. Prove that $KN = KP$.

2017 Oral Moscow Geometry Olympiad grades 8-9 p2
An isosceles trapezoid $ABCD$ with bases $BC$ and $AD$ is given. Circles with centers $O_1$ and $O_2$ are inscribed in triangles $ABC$ and $ABD$. Prove that direct $O_1O_2$ is perpendicular on $BC$.

2017 Oral Moscow Geometry Olympiad grades 8-9 p3
Points $M$ and $N$ are the midpoints of sides $AB$ and $CD$, respectively of quadrilateral $ABCD$. It is known that $BC // AD$ and $AN = CM$. Is it true that $ABCD$ is parallelogram?

2017 Oral Moscow Geometry Olympiad grades 8-9 p4
We consider triangles $ABC$, in which the point $M$ lies on the side $AB$, $AM = a, BM = b, CM = c$ ($c <a, c <b$). Find the smallest radius of the circumcircle of such triangles.

2017 Oral Moscow Geometry Olympiad grades 8-9 p5
Two squares are arranged as shown.
Prove that the area of ​​the black triangle equal to the sum of the gray areas.

Around triangle $ABC$ with acute angle C is circumscribed a circle. On the arc $AB$, which does not contain point $C$, point $D$ is chosen. Point $D'$ is symmetric on point $D$ with respect to line $AB$. Straight lines $AD'$ and $BD'$ intersect segments $BC$ and $AC$ at  points $E$ and $F$. Let point $C$ move along its arc $AB$. Prove that the center of the circumscribed circle of a triangle $CEF$ moves on a straight line.

2017 Oral Moscow Geometry Olympiad grades 10-11 p1
One square is inscribed in a circle, and another square is circumscribed around the same circle so that its vertices lie on the continuations of the sides of the first (see figure). Find the angle between the sides of these squares
2017 Oral Moscow Geometry Olympiad grades 10-11 p2
Given pyramid with base $n-gon$. How many maximum number of edges can be perpendicular to base?

2017 Oral Moscow Geometry Olympiad grades 10-11 p3
On the plane, a non-isosceles triangle is given, a circle circumscribed around it and the center of its inscribed circle are marked. Using only a ruler without tick marks and drawing no more than seven lines, build the diameter of the circumcircle.

2017 Oral Moscow Geometry Olympiad grades 10-11 p4
Prove that a circle constructed with the side $AB$ of a triangle $ABC$ as a diameter touches the inscribed circle of the triangle $ABC$ if and only if the side $AB$ is equal to the radius of the exircle on that side.

2017 Oral Moscow Geometry Olympiad grades 10-11 p5
The inscribed circle of the non-isosceles  triangle $ABC$ touches sides $AB, BC$ and $AC$ at points $C_1, A_1$ and $B_1$, respectively. The circumscribed circle of the triangle $A_1BC_1$ intersects the lines $B_1A_1$ and $B_1C_1$ at the points $A_0$ and $C_0$, respectively. Prove that the orthocenter of triangle $A_0BC_0$, the center of the inscribed circle of triangle $ABC$ and the midpoint  of the $AC$ lie on one straight line.

2017 Oral Moscow Geometry Olympiad grades 10-11 p6
Given acute angled traingle $ABC$ and altitudes $AA_1$, $BB_1$, $CC_1$.  Let $M$ midpoint of $BC$. $P$ point of intersection of circles $(AB_1C_1)$ and $(ABC)$ . $T$ is point of intersection of tangents to $(ABC)$ at $B$ and $C$. $S$ point of intersection of $AT$ and $(ABC)$. Prove that $P,A_1,S$ and midpoint of $MT$ collinear.



2018 Oral Moscow Geometry Olympiad grades 8-9 p1
Two parallelograms are arranged so as it shown on the picture. Prove that the diagonal of the one parallelogram passes through the intersection point of the other diagonal pour.

2018 Oral Moscow Geometry Olympiad grades 8-9 p2
Bisectors of angle $C$ and externalangle $A$ of trapezoid $ABCD$ with bases $BC$ and $AD$ intersect at point $M$, and the bisector of angle $B$ and external angle $D$ intersect at point $N$. Prove that the midpoint of the segment $MN$ is equidistant from the lines $AB$ and $CD$.

2018 Oral Moscow Geometry Olympiad grades 8-9 p3
On the extensions of sides $CA$ and $AB$ of triangle $ABC$ beyond points $A$ and $B$, respectively, the segments $AE = BC$ and $BF = AC$ are drawn. A circle is tangent to segment $BF$ at point $N$,  side $BC$ and the extension of side $AC$ beyond point $C$. Point $M$ is the midpoint of segment $EF$. Prove that the line $MN$ is parallel to the bisector of angle $A$.

2018 Oral Moscow Geometry Olympiad grades 8-9 p4
Given a triangle $ABC$ ($AB> AC$) and a circle circumscribed around it. Construct with a compass and a ruler the midpoint of the arc $BC$ (not containing vertex $A$), with no more than two lines (straight or circles).

2018 Oral Moscow Geometry Olympiad grades 8-9 p5
The circle circumscribed about an acute triangle $ABC$ and the vertex $C$ are fixed. Orthocenter $H$ moves in a circle with center at point $C$. Find the locus of the midpoints of the segments connecting the bases of heights drawn from vertices $A$ and $B$.

Cut each of the equilateral triangles with sides $2$ and $3$ into three parts and construct an equilateral triangle from all received parts.

2018 Oral Moscow Geometry Olympiad grades 10-11 p1
In a right triangle $ABC$ with a right angle $C$,  let  $AK$ and $BN$ be the angle bisectors. Let $D,E$ be the projections of $C$ on $AK, BN$ respectively. Prove that the length of the segment $DE$ is equal to the radius of the inscribed circle.

2018 Oral Moscow Geometry Olympiad grades 10-11 p2
The diagonals of the trapezoid $ABCD$  are perpendicular ($AD//BC, AD>BC$) . Point $M$ is the midpoint of the side of $AB$, the point $N$ is symmetric of  the center of the circumscribed circle of the triangle $ABD$  wrt  $AD$. Prove that $\angle CMN = 90^o$.

(A. Mudgal, India)
2018 Oral Moscow Geometry Olympiad grades 10-11 p3
A circle is fixed, point $A$ is on it and point $K$ outside the circle. The secant passing through $K$ intersects circle at points $P$ and $Q$. Prove that the orthocenters of the triangle $APQ$ lie on a fixed circle.

2018 Oral Moscow Geometry Olympiad grades 10-11 p4
On the side $AB$  of the triangle $ABC$, point $M$ is selected. In triangle $ACM$ point $I_1$ is the center of the inscribed circle, $J_1$ is the center of excircle wrt side $CM$. In the triangle $BCM$ point $I_2$ is the center of the inscribed cirlce, $J_2$ is the center of excircle wrt side $CM$. Prove that the line passing through the midpoints of the segments $I_1I_2$ and $J_1J_2$ is perpendicular to $AB$.

2018 Oral Moscow Geometry Olympiad grades 10-11 p5
Two ants sit on the surface of a tetrahedron. Prove that they can meet by breaking the sum of a distance not exceeding the diameter of a circle is circumscribed around the edge of a tetrahedron.

Let $ABC$ be an acute-angled triangle with circumcenter $O$. The circumcircle of $\triangle{BOC}$ meets the  lines $AB, AC$ at points $A_1, A_2$, respectively. Let $\omega_{A}$ be the circumcircle of triangle $AA_1A_2$. Define $\omega_B$ and $\omega_C$ analogously. Prove that the circles $\omega_A, \omega_B, \omega_C$ concur on $\odot(ABC)$.



2019 Oral Moscow Geometry Olympiad grades 8-9 p1
In the triangle $ABC, I$ is the center of the inscribed circle, point $M$ lies on the side of $BC$, with $\angle BIM = 90^o$. Prove that the distance from point $M$ to line $AB$ is equal to the diameter of the circle inscribed in triangle $ABC$

2019 Oral Moscow Geometry Olympiad grades 8-9 p2
On the side $AC$  of the triangle $ABC$ in the external side is constructed the parallelogram $ACDE$ . Let $O$ be the intersection point of its diagonals, $N$ and $K$ be  midpoints of BC and BA respectively. Prove that lines $DK, EN$ and $BO$ intersect at one point.

2019 Oral Moscow Geometry Olympiad grades 8-9 p3
In the acute triangle $ABC, \angle ABC = 60^o , O$ is the center of the circumscribed circle and  $H$ is the orthocenter. The bisector $BL$ intersects the circumscribed circle at the point $W, X$ is the intersection point of segments $WH$ and $AC$ . Prove that points $O, L, X$ and $H$ lie on the same circle.

2019 Oral Moscow Geometry Olympiad grades 8-9 p4
The perpendicular bisector of  the bisector $BL$ of the triangle $ABC$ intersects the bisectors of its external angles $A$ and $C$ at points $P$ and $Q$, respectively. Prove that the circle circumscribed around the triangle $PBQ$ is tangent to the circle circumscribed around the triangle $ABC$.

Given the segment$ PQ$ and a circle . A chord $AB$ moves around the circle, equal to $PQ$. Let $T$ be the intersection point of the perpendicular bisectors of the segments $AP$ and $BQ$. Prove that all points of $T$ thus obtained lie on one line.

2019 Oral Moscow Geometry Olympiad grades 8-9 p6
In the acute triangle $ABC$, the point $I_c$ is the center of excircle on the side $AB$, $A_1$ and $B_1$ are the tangency points of the other two excircles with sides $BC$ and $CA$, respectively, $C'$  is the point on the circumcircle diametrically opposite to point $C$. Prove that the lines $I_cC'$ and $A_1B_1$ are perpendicular.

2019 Oral Moscow Geometry Olympiad grades 10-11 p1
Circle inscribed in square $ABCD$ , is tangent to sides $AB$ and $CD$ at points $M$ and $K$ respectively. Line $BK$ intersects this circle at the point $L, X$ is the midpoint of $KL$. Find the angle $\angle MXK $.

2019 Oral Moscow Geometry Olympiad grades 10-11 p2
The angles of one quadrilateral are equal to the angles another quadrilateral. In addition, the corresponding angles between their diagonals are equal. Are these quadrilaterals necessarily similar?

2019 Oral Moscow Geometry Olympiad grades 10-11 p3
Restore the acute triangle $ABC$ given the vertex $A$, the base of the altitude drawn from the vertex $B$ and the center of the circle circumscribed around triangle $BHC$ (point $H$ is the orthocenter of triangle $ABC$).

2019 Oral Moscow Geometry Olympiad grades 10-11 p4
Given a right triangle $ABC$ ($\angle C=90^o$). The $C$-excircle touches the hypotenuse $AB$ at point $C_1, A_1$ is the touchpoint  of $B$-excircle with  line $BC, B_1$  is the touchpoint  of $A$-excircle with line $AC$. Find the angle $\angle A_1C_1B_1$.

2019 Oral Moscow Geometry Olympiad grades 10-11 p5
On sides $AB$ and $BC$ of a non-isosceles triangle $ABC$ are selected  points $C_1$ and $A_1$ such that the quadrilateral $AC_1A_1C$ is cyclic. Lines $CC_1$ and $AA_1$ intersect  at point $P$. Line $BP$ intersects the circumscribed circle of triangle $ABC$ at the point $Q$. Prove that the lines $QC_1$ and $CM$, where $M$ is the midpoint of $A_1C_1$, intersect at the circumscribed circles of triangle $ABC$.

The sum of the cosines of the flat angles of the trihedral angle is $-1$. Find the sum of its dihedral angles.


Points $A,B,C,D$ have been marked on checkered paper (see fig.). Find the tangent of the angle $ABD$.
A trapezoid is given in which one base is twice as large as the other. Use one ruler (no divisions) to draw the midline of this trapezoid

$ABCD$ is a convex quadrilateral such that $\angle A = \angle C < 90^{\circ}$ and $\angle ABD = 90^{\circ}$. $M$ is the midpoint of $AC$. Prove that $MB$ is perpendicular to $CD$.

On the diagonal $AC$ of cyclic quadrilateral $ABCD$ a point $E$ is chosen such that $\angle ABE = \angle CBD$. Points $O,O_1,O_2$ are the circumcircles of triangles $ABC, ABE$ and $CBE$ respectively. Prove that lines $DO,AO_{1}$ and $CO_{2}$ are concurrent.

The trapezoid is inscribed in a circle. Prove that the sum of distances from any point of the circle to the midpoints of the lateral sides are not less than the diagonal of the trapezoid.

Point $M$ is a midpoint of side $BC$ of a triangle $ABC$ and $H$ is the orthocenter of $ABC$. $MH$ intersects the $A$-angle bisector at $Q$. Points $X$ and $Y$ are the projections of $Q$ on sides $AB$ and $AC$. Prove that $XY$ passes through $H$.

Quadrilateral $ABCD$ is inscribed in a circle, $E$ is an arbitrary point of this circle. It is known that distances from point $E$ to lines $AB, AC, BD$ and $CD$ are equal to $a, b, c$ and $d$ respectively. Prove that $ad= bc$.

Two quadrangles have equal areas, perimeters and corresponding angles. Are such quadrilaterals necessarily congurent ?

Circle $(O)$ and its chord $BC$ are given. Point $A$ moves on the major arc $BC$. $AL$ is the angle bisector in a triangle $ABC$. Show that the disctance from the circumcenter of triangle $AOL$ to the line $BC$ does not depend on the position of point $A$.

Points $STABCD$ in space form a convex octahedron with faces $SAB,SBC,SCD,SDA,TAB,TBC,TCD,TDA$ such that there exists a sphere that is tangent to all of its edges. Prove that $A,B,C,D$ lie in one plane.

Let $ABC$ be a triangle, $I$ and $O$ be its incenter and circumcenter respectively. $A'$ is symmetric to $O$ with respect to line $AI$. Points $B'$ and $C'$ are defined similarly. Prove that the nine-point centers of triangles $ABC$ and $A'B'C'$ coincide.

$ABCD$ is a square and $XYZ$ is an equilateral triangle such that $X$ lies on $AB$, $Y$ lies on $BC$ and $Z$ lies on $DA$. Line through the centers of $ABCD$ and $XYZ$ intersects $CD$ at $T$. Find angle $CTY$


Given an isosceles trapezoid $ABCD$. The bisector of angle $B$ intersects the base $AD$ at point $L$. Prove that the center of the circle circumscribed around triangle $BLD$ lies on the circle circumscribed around the trapezoid.

Angle bisectors from vertices $B$ and $C$ and the perpendicular bisector of side $BC$ are drawn in a non-isosceles triangle $ABC$. Next, three points of pairwise intersection of these three lines were marked (remembering which point is which), and the triangle itself was erased. Restore it according to the marked points using a compass and ruler.

In quadrilateral $ABCD$, sides $AB$ and $CD$ are equal (but not parallel), points $M$ and $N$ are the midpoints of $AD$ and $BC$. The perpendicular bisector of $MN$ intersects sides $AB$ and $CD$ at points $P$ and $Q$, respectively. Prove that $AP = CQ$.

In triangle $ABC$, angle $C$ is equal to $60^o$. Bisectors $AA'$ and $BB'$ intersect at point $I$. Point $K$ is symmetric to $I$ with respect to line $AB$. Prove that lines $CK$ and $A'B'$ are perpendicular.

Given a circle and a straight line $AB$ passing through its center (points $A$ and $B$ are fixed, $A$ is outside the circle, and $B$ is inside). Find the locus of the intersection of lines $AX$ and $BY$, where $XY$ is an arbitrary diameter of the circle.

In an acute non-isosceles triangle $ABC$, the inscribed circle touches side $BC$ at point $T, Q$ is the midpoint of altitude $AK$, $P$ is the orthocenter of the triangle formed by the bisectors of angles $B$ and $C$ and line $AK$. Prove that the points $P, Q$ and $T$ lie on the same line.


In a circle with center $O$, chords $AB$ and $AC$ are drawn, both equal to the radius. Points $A_1$, $B_1$ and $C_1$ are projections of points $A, B$ and $C$, respectively, onto an arbitrary diameter $XY$. Prove that one of the segments $XB_1$, $OA_1$ and $C_1Y$ is equal to the sum of the other two.

In an acute triangle $ABC$,$O$ is the center of the circumscribed circle $\omega$, $P$ is the point of intersection of the tangents to $\omega$ through the points $B$ and $C$, the median AM intersects the circle $\omega$ at point $D$. Prove that points $A, D, P$ and $O$ lie on the same circle.

Extensions of opposite sides of a convex quadrilateral $ABCD$ intersect at points $P$ and $Q$. Points are marked on the sides of $ABCD$ (one per side), which are the vertices of a parallelogram with a side parallel to $PQ$. Prove that the intersection point of the diagonals of this parallelogram lies on one of the diagonals of quadrilateral $ABCD$.
An acute-angled non-isosceles triangle $ABC$ is drawn, a circumscribed circle and its center $O$ are drawn. The midpoint of side $AB$ is also marked. Using only a ruler (no divisions), construct the triangle's orthocenter by drawing no more than $6$ lines.

Circle $\omega$ is tangent to the interior of the circle $\Omega$ at the point C. Chord $AB$ of circle $\Omega$ is tangent to $\omega$. Chords $CF$ and $BG$ of circle $\Omega$ intersect at point $E$ lying on $\omega$. Prove that the circumcircle of triangle $CGE$ is tangent to straight line $AF$.

In a tetrahedron, segments connecting the midpoints of heights with the orthocenters of the faces to which these heights are drawn intersect at one point. Prove that in such a tetrahedron all faces are equal or there are perpendicular edges.




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