SHARYGIN 2023

Let $L$ be the midpoint of the minor arc $AC$ of the circumcircle of an acute-angled triangle $ABC$ . A point $P$ is the projection of $B$ to the tangent at $L$ to the circumcircle. Prove that $P$ , $L$ , and the midpoints of sides $AB$ , $BC$ are concyclic.

The diagonals of a rectangle $ABCD$ meet at point $E$ . A circle centered at $E$ lies inside the rectangle. Let $CF$ , $DG$ , $AH$ be the tangents to this circle from $C$ , $D$ , $A$ ; let $CF$ meet $DG$ at point $I$ , $EI$ meet $AD$ at point $J$ , and $AH$ meet $CF$ at point $L$ . Prove that $LJ$ is perpendicular to $AD$ .

A circle touches the lateral sides of a trapezoid $ABCD$ at points $B$ and $C$ , and its center lies on $AD$ . Prove that the diameter of the circle is less than the medial line of the trapezoid.

Points $D$ and $E$ lie on the lateral sides $AB$ and $BC$ respectively of an isosceles triangle $ABC$ in such a way that $\angle BED = 3\angle BDE$ . Let $D'$ be the reflection of $D$ about $AC$ . Prove that the line $D'E$ passes through the incenter of $ABC$ .

Let $ABCD$ be a cyclic quadrilateral. Points $E$ and $F$ lie on the sides $AD$ and $CD$ in such a way that $AE = BC$ and $AB = CF$ . Let $M$ be the midpoint of $EF$ . Prove that $\angle AMC = 90^{\circ}$ .

Let $A_1, B_1, C_1$ be the feet of altitudes of an acute-angled triangle $ABC$ . The incircle of triangle $A_1B_1C_1$ touches $A_1B_1, A_1C_1, B_1C_1$ at points $C_2, B_2, A_2$ respectively. Prove that the lines $AA_2, BB_2, CC_2$ concur at a point lying on the Euler line of triangle $ABC$ .

Let $A$ be a fixed point of a circle $\omega$ . Let $BC$ be an arbitrary chord of $\omega$ passing through a fixed point $P$ . Prove that the nine-points circles of triangles $ABC$ touch some fixed circle not depending on $BC$ .

A triangle $ABC$ $(a>b>c)$ is given. Its incenter $I$ and the touching points $K, N$ of the incircle with $BC$ and $AC$ respectively are marked. Construct a segment with length $a-c$ using only a ruler and drawing at most three lines.

It is known that the reflection of the orthocenter of a triangle $ABC$ about its circumcenter lies on $BC$ . Let $A_1$ be the foot of the altitude from $A$ . Prove that $A_1$ lies on the circle passing through the midpoints of the altitudes of $ABC$ .

Altitudes $BE$ and $CF$ of an acute-angled triangle $ABC$ meet at point $H$ . The perpendicular from $H$ to $EF$ meets the line $\ell$ passing through $A$ and parallel to $BC$ at point $P$ . The bisectors of two angles between $\ell$ and $HP$ meet $BC$ at points $S$ and $T$ . Prove that the circumcircles of triangles $ABC$ and $PST$ are tangent.

Let $H$ be the orthocenter of an acute-angled triangle $ABC$ ; $E$ , $F$ be points on $AB, AC$ respectively, such that $AEHF$ is a parallelogram; $X, Y$ be the common points of the line $EF$ and the circumcircle $\omega$ of triangle $ABC$ ; $Z$ be the point of $\omega$ opposite to $A$ . Prove that $H$ is the orthocenter of triangle $XYZ$ .

Let $ABC$ be a triangle with obtuse angle $B$ , and $P, Q$ lie on $AC$ in such a way that $AP = PB, BQ = QC$ . The circle $BPQ$ meets the sides $AB$ and $BC$ at points $N$ and $M$ respectively. $\textbf{(a)}$ (grades 8-9) Prove that the distances from the common point $R$ of $PM$ and $NQ$ to $A$ and $C$ are equal. $\textbf{(b)}$ (grades 10-11) Let $BR$ meet $AC$ at point $S$ . Prove that $MN \perp OS$ , where $O$ is the circumcenter of $ABC$ .

The base $AD$ of a trapezoid $ABCD$ is twice greater than the base $BC$ , and the angle $C$ equals one and a half of the angle $A$ . The diagonal $AC$ divides angle $C$ into two angles. Which of them is greater?

Suppose that a closed oriented polygonal line $\mathcal{L}$ in the plane does not pass through a point $O$ , and is symmetric with respect to $O$ . Prove that the winding number of $\mathcal{L}$ around $O$ is odd.

The winding number of $\mathcal{L}$ around $O$ is defined to be the following sum of the oriented angles divided by $2\pi$ : $$\deg_O\mathcal{L} := \dfrac{\angle A_1OA_2+\angle A_2OA_3+\dots+\angle A_{n-1}OA_n+\angle A_nOA_1}{2\pi}.$$

Let $ABCD$ be a convex quadrilateral. Points $X$ and $Y$ lie on the extensions beyond $D$ of the sides $CD$ and $AD$ respectively in such a way that $DX = AB$ and $DY = BC$ . Similarly points $Z$ and $T$ lie on the extensions beyond $B$ of the sides $CB$ and $AB$ respectively in such a way that $BZ = AD$ and $BT = DC$ . Let $M_1$ be the midpoint of $XY$ , and $M_2$ be the midpoint of $ZT$ . Prove that the lines $DM_1, BM_2$ and $AC$ concur.

Let $AH_A$ and $BH_B$ be the altitudes of a triangle $ABC$ . The line $H_AH_B$ meets the circumcircle of $ABC$ at points $P$ and $Q$ . Let $A'$ be the reflection of $A$ about $BC$ , and $B'$ be the reflection of $B$ about $CA$ . Prove that $A',B', P,Q$ are concyclic.

A common external tangent to circles $\omega_1$ and $\omega_2$ touches them at points $T_1, T_2$ respectively. Let $A$ be an arbitrary point on the extension of $T_1T_2$ beyond $T_1$ , and $B$ be a point on the extension of $T_1T_2$ beyond $T_2$ such that $AT_1 = BT_2$ . The tangents from $A$ to $\omega_1$ and from $B$ to $\omega_2$ distinct from $T_1T_2$ meet at point $C$ . Prove that all nagelians of triangles $ABC$ from $C$ have a common point.

Restore a bicentral quadrilateral $ABCD$ if the midpoints of the arcs $AB,BC,CD$ of its circumcircle are given.

A cyclic quadrilateral $ABCD$ is given. An arbitrary circle passing through $C$ and $D$ meets $AC,BC$ at points $X,Y$ respectively. Find the locus of common points of circles $CAY$ and $CBX$ .

Let a point $D$ lie on the median $AM$ of a triangle $ABC$ . The tangents to the circumcircle of triangle $BDC$ at points $B$ and $C$ meet at point $K$ . Prove that $DD'$ is parallel to $AK$ , where $D'$ is isogonally conjugated to $D$ with respect to $ABC$ .

Let $ABCD$ be a cyclic quadrilateral; $M_{ac}$ be the midpoint of $AC$ ; $H_d,H_b$ be the orthocenters of $\triangle ABC,\triangle ADC$ respectively; $P_d,P_b$ be the projections of $H_d$ and $H_b$ to $BM_{ac}$ and $DM_{ac}$ respectively. Define similarly $P_a,P_c$ for the diagonal $BD$ . Prove that $P_a,P_b,P_c,P_d$ are concyclic.

Let $ABC$ be a scalene triangle, $M$ be the midpoint of $BC,P$ be the common point of $AM$ and the incircle of $ABC$ closest to $A$ , and $Q$ be the common point of the ray $AM$ and the excircle farthest from $A$ . The tangent to the incircle at $P$ meets $BC$ at point $X$ , and the tangent to the excircle at $Q$ meets $BC$ at $Y$ . Prove that $MX=MY$ .

An ellipse $\Gamma_1$ with foci at the midpoints of sides $AB$ and $AC$ of a triangle $ABC$ passes through $A$ , and an ellipse $\Gamma_2$ with foci at the midpoints of $AC$ and $BC$ passes through $C$ . Prove that the common points of these ellipses and the orthocenter of triangle $ABC$ are collinear.

A tetrahedron $ABCD$ is give. A line $\ell$ meets the planes $ABC,BCD,CDA,DAB$ at points $D_0,A_0,B_0,C_0$ respectively. Let $P$ be an arbitrary point not lying on $\ell$ and the planes of the faces, and $A_1,B_1,C_1,D_1$ be the second common points of lines $PA_0,PB_0,PC_0,PD_0$ with the spheres $PBCD,PCDA,PDAB,PABC$ respectively. Prove $P,A_1,B_1,C_1,D_1$ lie on a circle.