SHARYGIN 2020

Let $ABC$ be a triangle with $\angle C=90^\circ$ , and $A_0$ , $B_0$ , $C_0$ be the mid-points of sides $BC$ , $CA$ , $AB$ respectively. Two regular triangles $AB_0C_1$ and $BA_0C_2$ are constructed outside $ABC$ . Find the angle $C_0C_1C_2$ .

Let $ABCD$ be a cyclic quadrilateral. A circle passing through $A$ and $B$ meets $AC$ and $BD$ at points $E$ and $F$ respectively. The lines $AF$ and $BC$ meet at point $P$ , and the lines $BE$ and $AD$ meet at point $Q$ . Prove that $PQ$ is parallel to $CD$ .

Let $ABC$ be a triangle with $\angle C=90^\circ$ , and $D$ be a point outside $ABC$ , such that $\angle ADC=\angle BAC$ . The segments $CD$ and $AB$ meet at point $E$ . It is known that the distance from $E$ to $AC$ is equal to the circumradius of triangle $ADE$ . Find the angles of triangle $ABC$ .

Let $ABCD$ be an isosceles trapezoid with bases $AB$ and $CD$ . Prove that the centroid of triangle $ABD$ lies on $CF$ where $F$ is the projection of $D$ to $AB$ .

Let $BB_1$ , $CC_1$ be the altitudes of triangle $ABC$ , and $AD$ be the diameter of its circumcircle. The lines $BB_1$ and $DC_1$ meet at point $E$ , the lines $CC_1$ and $DB_1$ meet at point $F$ . Prove that $\angle CAE = \angle BAF$ .

Circles $\omega_1$ and $\omega_2$ meet at point $P,Q$ . Let $O$ be the common point of external tangents of $\omega_1$ and $\omega_2$ . A line passing through $O$ meets $\omega_1$ and $\omega_2$ at points $A,B$ located on the same side with respect to line segment $PQ$ .The line $PA$ meets $\omega_2$ for the second time at $C$ and the line $QB$ meets $\omega_1$ for the second time at $D$ . Prove that $O-C-D$ are collinear.

Prove that the medial lines of triangle $ABC$ meets the sides of triangle formed by its excenters at six concyclic points.

Two circles meeting at points $P$ and $R$ are given. Let $\ell_1$ , $\ell_2$ be two lines passing through $P$ . The line $\ell_1$ meets the circles for the second time at points $A_1$ and $B_1$ . The tangents at these points to the circumcircle of triangle $A_1RB_1$ meet at point $C_1$ . The line $C_1R$ meets $A_1B_1$ at point $D_1$ . Points $A_2$ , $B_2, C_2, D_2$ are defined similarly. Prove that the circles $D_1D_2P$ and $C_1C_2R$ touch.

The vertex $A$ , center $O$ and Euler line $\ell$ of a triangle $ABC$ is given. It is known that $\ell$ intersects $AB,AC$ at two points equidistant from $A$ . Restore the triangle.

Given are a closed broken line $A_1A_2\ldots A_n$ and a circle $\omega$ which touches each of lines $A_1A_2,A_2A_3,\ldots,A_nA_1$ . Call the link good, if it touches $\omega$ , and bad otherwise (i.e. if the extension of this link touches $\omega$ ). Prove that the number of bad links is even.

Let $ABC$ be a triangle with $\angle A=60^{\circ}$ , $AD$ be its bisector, and $PDQ$ be a regular triangle with altitude $DA$ . The lines $PB$ and $QC$ meet at point $K$ . Prove that $AK$ is a symmedian of $ABC$ .

Let $H$ be the orthocenter of a nonisosceles triangle $ABC$ . The bisector of angle $BHC$ meets $AB$ and $AC$ at points $P$ and $Q$ respectively. The perpendiculars to $AB$ and $AC$ from $P$ and $Q$ meet at $K$ . Prove that $KH$ bisects the segment $BC$ .

Let $I$ be the incenter of triangle $ABC$ . The excircle with center $I_A$ touches the side $BC$ at point $A'$ . The line $l$ passing through $I$ and perpendicular to $BI$ meets $I_AA'$ at point $K$ lying on the medial line parallel to $BC$ . Prove that $\angle B \leq 60^\circ$ .

A non-isosceles triangle is given. Prove that one of the circles touching internally its incircle and circumcircle and externally one of its excircles passes through a vertex of the triangle.

A circle passing through the vertices $B$ and $D$ of quadrilateral $ABCD$ meets $AB$ , $BC$ , $CD$ , and $DA$ at points $K$ , $L$ , $M$ , and $N$ respectively. A circle passing through $K$ and $M$ meets $AC$ at $P$ and $Q$ . Prove that $L$ , $N$ , $P$ , and $Q$ are concyclic.

Cevians $AP$ and $AQ$ of a triangle $ABC$ are symmetric with respect to its bisector. Let $X$ , $Y$ be the projections of $B$ to $AP$ and $AQ$ respectively, and $N$ , $M$ be the projections of $C$ to $AP$ and $AQ$ respectively. Prove that $XM$ and $NY$ meet on $BC$ .

Chords $A_1A_2$ and $B_1B_2$ meet at point $D$ . Suppose $D'$ is the inversion image of $D$ and the line $A_1B_1$ meets the perpendicular bisector to $DD'$ at a point $C$ . Prove that $CD\parallel A_2B_2$ .

Bisectors $AA_1$ , $BB_1$ , and $CC_1$ of triangle $ABC$ meet at point $I$ . The perpendicular bisector to $BB_1$ meets $AA_1,CC_1$ at points $A_0,C_0$ respectively. Prove that the circumcircles of triangles $A_0IC_0$ and $ABC$ touch.

Quadrilateral $ABCD$ is such that $AB \perp CD$ and $AD \perp BC$ . Prove that there exist a point such that the distances from it to the sidelines are proportional to the lengths of the corresponding sides.

The line touching the incircle of triangle $ABC$ and parallel to $BC$ meets the external bisector of angle $A$ at point $X$ . Let $Y$ be the midpoint of arc $BAC$ of the circumcircle. Prove that the angle $XIY$ is right.

The diagonals of bicentric quadrilateral $ABCD$ meet at point $L$ . Given are three segments equal to $AL$ , $BL$ , $CL$ . Restore the quadrilateral using a compass and a ruler.

Let $\Omega$ be the circumcircle of cyclic quadrilateral $ABCD$ . Consider such pairs of points $P$ , $Q$ of diagonal $AC$ that the rays $BP$ and $BQ$ are symmetric with respect the bisector of angle $B$ . Find the locus of circumcenters of triangles $PDQ$ .

A non-self-intersecting polygon is nearly convex if precisely one of its interior angles is greater than $180^\circ$ .

One million distinct points lie in the plane in such a way that no three of them are collinear. We would like to construct a nearly convex one-million-gon whose vertices are precisely the one million given points. Is it possible that there exist precisely ten such polygons?

Let $I$ be the incenter of a tetrahedron $ABCD$ , and $J$ be the center of the exsphere touching the face $BCD$ containing three remaining faces (outside these faces). The segment $IJ$ meets the circumsphere of the tetrahedron at point $K$ . Which of two segments $IJ$ and $JK$ is longer?