SHARYGIN 2016

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.

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 $ABCDEF$ 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 $BC, CA, AB$ respectively of a non-isosceles triangle $ABC$ . Points $H_A, H_B, H_C$ lie on the corresponding sides, different from $M_A, M_B, M_C$ such that $M_AH_B=M_AH_C,$ $M_BH_A=M_BH_C,$ and $M_CH_A=M_CH_B$ . Prove that $H_A, H_B, H_C$ are the feet of the corresponding altitudes.

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 $\angle A_oBB' = \angle B_oCC' =\angle C_oAA'$ . a) Find all possible values of these angles. b) Prove that segments $AA_o, BB_o, CC_o$ concur. c) Prove that the projections of the incenter to $A'B', B'C', C'A'$ are the vertices of a regular triangle.