SHARYGIN 2017

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 $C$ . Proposed 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 $C$ . Proposed 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? Proposed 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 $ABC$ . Proposed 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. Proposed 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. Proposed 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?