SHARYGIN 2013

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 $CK$ is parallel to $AB$ .

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.

Chords $BC$ and $DE$ of circle $\omega$ meet at point $A$ . The line through $D$ parallel to $BC$ meets $\omega$ again at $F$ , and $FA$ meets $\omega$ again at $T$ . Let $M = ET \cap BC$ and let $N$ be the reflection of $A$ over $M$ . Show that $(DEN)$ passes through the midpoint of $BC$ .

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?