SHARYGIN 2019

Let $A_1A_2A_3$ be an acute-angled triangle inscribed into a unit circle centered at $O$ . The cevians from $A_i$ passing through $O$ meet the opposite sides at points $B_i$ $(i = 1, 2, 3)$ respectively. [list=a] [] Find the minimal possible length of the longest of three segments $B_iO$ . [] Find the maximal possible length of the shortest of three segments $B_iO$ . [/list]

Let $ABC$ be an acute-angled triangle with altitude $AT = h$ . The line passing through its circumcenter $O$ and incenter $I$ meets the sides $AB$ and $AC$ at points $F$ and $N$ , respectively. It is known that $BFNC$ is a cyclic quadrilateral. Find the sum of the distances from the orthocenter of $ABC$ to its vertices.

Let the side $AC$ of triangle $ABC$ touch the incircle and the corresponding excircle at points $K$ and $L$ respectively. Let $P$ be the projection of the incenter onto the perpendicular bisector of $AC$ . It is known that the tangents to the circumcircle of triangle $BKL$ at $K$ and $L$ meet on the circumcircle of $ABC$ . Prove that the lines $AB$ and $BC$ touch the circumcircle of triangle $PKL$ .

The incircle $\omega$ of triangle $ABC$ touches the sides $BC$ , $CA$ and $AB$ at points $D$ , $E$ and $F$ respectively. The perpendicular from $E$ to $DF$ meets $BC$ at point $X$ , and the perpendicular from $F$ to $DE$ meets $BC$ at point $Y$ . The segment $AD$ meets $\omega$ for the second time at point $Z$ . Prove that the circumcircle of the triangle $XYZ$ touches $\omega$ .

Let $AH_1$ and $BH_2$ be the altitudes of triangle $ABC$ . Let the tangent to the circumcircle of $ABC$ at $A$ meet $BC$ at point $S_1$ , and the tangent at $B$ meet $AC$ at point $S_2$ . Let $T_1$ and $T_2$ be the midpoints of $AS_1$ and $BS_2$ respectively. Prove that $T_1T_2$ , $AB$ and $H_1H_2$ concur.

Three circles $\omega_1$ , $\omega_2$ , $\omega_3$ are given. Let $A_0$ and $A_1$ be the common points of $\omega_1$ and $\omega_2$ , $B_0$ and $B_1$ be the common points of $\omega_2$ and $\omega_3$ , $C_0$ and $C_1$ be the common points of $\omega_3$ and $\omega_1$ . Let $O_{i,j,k}$ be the circumcenter of triangle $A_iB_jC_k$ . Prove that the four lines of the form $O_{ijk}O_{1 - i,1 - j,1 - k}$ are concurrent or parallel.

A quadrilateral $ABCD$ without parallel sidelines is circumscribed around a circle centered at $I$ . Let $K, L, M$ and $N$ be the midpoints of $AB, BC, CD$ and $DA$ respectively. It is known that $AB \cdot CD = 4IK \cdot IM$ . Prove that $BC \cdot AD = 4IL \cdot IN$ .

Let $AL_a$ , $BL_b$ , $CL_c$ be the bisecors of triangle $ABC$ . The tangents to the circumcircle of $ABC$ at $B$ and $C$ meet at point $K_a$ , points $K_b$ , $K_c$ are defined similarly. Prove that the lines $K_aL_a$ , $K_bL_b$ and $K_cL_c$ concur.

Let $O$ be the circumcenter of triangle ABC, $H$ be its orthocenter, and $M$ be the midpoint of $AB$ . The line $MH$ meets the line passing through $O$ and parallel to $AB$ at point $K$ lying on the circumcircle of $ABC$ . Let $P$ be the projection of $K$ onto $AC$ . Prove that $PH \parallel BC$ .

An ellipse $\Gamma$ and its chord $AB$ are given. Find the locus of orthocenters of triangles $ABC$ inscribed into $\Gamma$ .

Let $AA_0$ be the altitude of the isosceles triangle $ABC~(AB = AC)$ . A circle $\gamma$ centered at the midpoint of $AA_0$ touches $AB$ and $AC$ . Let $X$ be an arbitrary point of line $BC$ . Prove that the tangents from $X$ to $\gamma$ cut congruent segments on lines $AB$ and $AC$

In the plane, let $a$ , $b$ be two closed broken lines (possibly self-intersecting), and $K$ , $L$ , $M$ , $N$ be four points. The vertices of $a$ , $b$ and the points $K$ $L$ , $M$ , $N$ are in general position (i.e. no three of these points are collinear, and no three segments between them concur at an interior point). Each of segments $KL$ and $MN$ meets $a$ at an even number of points, and each of segments $LM$ and $NK$ meets $a$ at an odd number of points. Conversely, each of segments $KL$ and $MN$ meets $b$ at an odd number of points, and each of segments $LM$ and $NK$ meets $b$ at an even number of points. Prove that $a$ and $b$ intersect.

Two unit cubes have a common center. Is it always possible to number the vertices of each cube from $1$ to $8$ so that the distance between each pair of identically numbered vertices would be at most $4/5$ ? What about at most $13/16$ ?