SHARYGIN 2010

Does there exist a triangle, whose side is equal to some of its altitudes, another side is equal to some of its bisectrices, and the third side 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 the triangle $CA_1B_1$. Prove that $OI \perp AB$.

Points $A', B', C'$ lie on sides $BC, CA, AB$ of a 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 the 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 into a circle centered at $N$.

A point $E$ lies on the altitude $BD$ of a triangle $ABC$, and $\angle AEC = 90^\circ$. Points $O_1$ and $O_2$ are the circumcenters of the 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 a regular triangle $ABC$ ($M$ is between $B$ and $N$), and $\angle MAN = 30^\circ$. The circumcircles of the triangles $AMC$ and $ANB$ meet at a point $K$. Prove that the line $AK$ passes through the circumcenter of the 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 $ABC$. Points $I_b$ and $I_c$ are the incenters of the triangles $ABH$ and $ACH$ respectively; $BC$ touches the incircle of $ABC$ 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 a 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 $AD/BD = BC/AC$.

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 $S_{ABCD} \geq 3S_{BCM}$.

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 the 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) $A_1N/BB_1 + A_1L/CC_1 = 1$.

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 a 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 drown 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 $V$. 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 $ABC$. 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 $AB \cdot CF = 2BC \cdot FA, CD \cdot EB = 2DE \cdot BC$, and $EF \cdot AD = 2FA \cdot DE$. Prove that the lines $AD, BE$ and $CF$ concur.

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