[Solutions] Sharygin Geometry Mathematical Olympiad 2020 (Correspondence Round)

1. 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 $C_0C_1C_2$. 
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 $CD$. 
3. 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 $ABC$. 
4. 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 $AB$. 
5. 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 $\angle CAE = \angle BAF$. 
6. 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. 
7. Prove that the medial lines of triangle $ABC$ meets the sides of triangle formed by its excenters at six concyclic points. 
8. 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. 
9. 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. 
10. 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. 
11. 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 $ABC$.
12. 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 $BC$. 
13. 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 $\angle B \leq 60^\circ$. 
14. 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.
15. 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. 
16. 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 $BC$. 
17. 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 $CD\parallel A_2B_2$. 
18. 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. 
19. 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. 
20. 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. 
21. 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. 
22. 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 $PDQ$. 
23. 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?
24. 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?

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