[Solutions] Sharygin Geometry Mathematical Olympiad 2016 (Correspondence Round)

1. A trapezoid $A B C D$ with bases $A D$ and $B C$ is such that $A B=B D$. Let $M$ be the midpoint of $D C .$ Prove that $\angle M B C=\angle B C A$.
2. Mark three nodes on a cellular paper so that the semiperimeter of the obtained triangle would be equal to the sum of its two smallest medians.
3. Let $A H_{1}, B H_{2}$ be two altitudes of an acute-angled triangle $A B C, D$ be the projection of $H_{1}$ to $A C, E$ be the projection of $D$ to $A B, F$ be the common point of $E D$ and $A H_{1}$ Prove that $H_{2} F \| B C$.
4. In quadrilateral $A B C D \angle B=\angle D=90^{\circ}$ and $A C=B C+D C$. Point $P$ of ray $B D$ is such that $B P=A D$. Prove that line $C P$ is parallel to the bisector of angle $A B D$.
5. In quadrilateral $A B C D A B=C D, M$ and $K$ are the midpoints of $B C$ and $A D$. Prove that the angle between $M K$ and $A C$ is equal to the half-sum of angles $B A C$ and $D C A$
6. Let $M$ be the midpoint of side $A C$ of triangle $A B C, M D$ and $M E$ be the perpendiculars from $M$ to $A B$ and $B C$ respectively. Prove that the distance between the circumcenters of triangles $A B E$ and $B C D$ is equal to $A C / 4$
7. Let all distances between the vertices of a convex $n$ -gon $(n>3)$ be different.
   a) A vertex is called uninteresting if the closest vertex is adjacent to it. What is the minimal possible number of uninteresting vertices (for a given $n$ )?
   b) A vertex is called unusual if the farthest vertex is adjacent to it. What is the maximal possible number of unusual vertices (for a given $n$ )?
8. Let $A B C D E$ be an inscribed pentagon such that $\angle B+\angle E=\angle C+\angle D$. Prove that $\angle C A D<\pi / 3<\angle A$
9. Let $A B C$ be a right-angled triangle and $C H$ be the altitude from its right angle $C .$ Points $O_{1}$ and $O_{2}$ are the incenters of triangles $A C H$ and $B C H$ respectively; $P_{1}$ and $P_{2}$ are the touching points of their incircles with $A C$ and $B C$. Prove that lines $O_{1} P_{1}$ and $O_{2} P_{2}$ meet on $A B$
10. Point $X$ moves along side $A B$ of triangle $A B C,$ and point $Y$ moves along its circumcircle in such a way that line $X Y$ passes through the midpoint of arc $A B .$ Find the locus of the circumcenters of triangles $I X Y$, where $I$ is the incenter of $A B C$.
11. Restore a triangle $A B C$ by vertex $B$, the centroid and the common point of the symmedian from $B$ with the circumcircle.
12. Let $B B_{1}$ be the symmedian of a nonisosceles acute-angled triangle $A B C$. Ray $B B_{1}$ meets the circumcircle of $A B C$ for the second time at point $L .$ Let $A H_{A}, B H_{B}, C H_{C}$ be the altitudes of triangle $A B C .$ Ray $B H_{B}$ meets the circumcircle of $A B C$ for the second time at point $T$. Prove that $H_{A}, H_{C}, T, L$ are concyclic.
13. Given are a triangle $A B C$ and a line $\ell$ meeting $B C, A C, A B$ at points $L_{a}, L_{b}$ $L_{c}$ respectively. The perpendicular from $L_{a}$ to $B C$ meets $A B$ and $A C$ at points $A_{B}$ and $A_{C}$ respectively. Point $O_{a}$ is the circumcenter of triangle $A A_{b} A_{c} .$ Points $O_{b}$ and $O_{c}$ are defined similarly. Prove that $O_{a}, O_{b}$ and $O_{c}$ are collinear.
14. Let a triangle $A B C$ be given. Consider the circle touching its circumcircle at $A$ and touching externally its incircle at some point $A_{1} .$ Points $B_{1}$ and $C_{1}$ are defined similarly.
    a) Prove that lines $A A_{1}, B B_{1}$ и $C C_{1}$ concur.
    b) Let $A_{2}$ be the touching point of the incircle with $B C .$ Prove that lines $A A_{1}$ and $A A_{2}$ are symmetric about the bisector of angle $A$.
15. Let $O, M, N$ be the circumcenter, the centroid and the Nagel point of a triangle. Prove that angle $M O N$ is right if and only if one of the triangle's angles is equal to $60^{\circ} .$
16. Let $B B_{1}$ and $C C_{1}$ be altitudes of triangle $A B C$. The tangents to the circumcircle of $A B_{1} C_{1}$ at $B_{1}$ and $C_{1}$ meet $A B$ and $A C$ at points $M$ and $N$ respectively. Prove that the common point of circles $A M N$ and $A B_{1} C_{1}$ distinct from $A$ lies on the Euler line of $A B C$
17. Let $D$ be an arbitrary point on side $B C$ of triangle $A B C .$ Circles $\omega_{1}$ and $\omega_{2}$ pass through $A$ and $D$ in such a way that $B A$ touches $\omega_{1}$ and $C A$ touches $\omega_{2}$. Let $B X$ be the second tangent from $B$ to $\omega_{1},$ and $C Y$ be the second tangent from $C$ to $\omega_{2} .$ Prove that the circumcircle of triangle $X D Y$ touches $B C$.
18. Let $A B C$ be a triangle with $\angle C=90^{\circ},$ and $K, L$ be the midpoints of the minor $\operatorname{arcs} A C$ and $B C$ of its circumcircle. Segment $K L$ meets $A C$ at point $N .$ Find angle $N I C$ where $I$ is the incenter of $A B C$.
19. Let $A B C D E F$ be a regular hexagon. Points $P$ and $Q$ on tangents to its circumcircle at $A$ and $D$ respectively are such that $P Q$ touches the minor arc $E F$ of this circle. Find the angle between $P B$ and $Q C$.
20. The incircle $\omega$ of a triangle $A B C$ touches $B C, A C$ and $A B$ at points $A_{0}, B_{0}$ and $C_{0}$ respectively. The bisectors of angles $B$ and $C$ meet the perpendicular bisector to segment $A A_{0}$ at points $Q$ and $P$ respectively. Prove that $P C_{0}$ and $Q B_{0}$ meet on $\omega$.
21. The areas of rectangles $P$ and $Q$ are equal, but the diagonal of $P$ is greater. Rectangle $Q$ can be covered by two copies of $P .$ Prove that $P$ can be covered by two copies of $Q$
22. Let $M_{A}, M_{B}, M_{C}$ be the midpoints of the sides of a nonisosceles triangle $A B C$. Points $H_{A}, H_{B}, H_{C}$ lying on the correspondent sides and distinct from $M_{A}, M_{B}, M_{C}$ are such that $M_{A} H_{B}=M_{A} H_{C}, M_{B} H_{A}=M_{B} H_{C}, M_{C} H_{A}=M_{C} H_{B} .$ Prove that $H_{A}, H_{B}, H_{C}$ are the bases of the altitudes of $A B C$.
23. A sphere touches all edges of a tetrahedron. Let $a, b, c$ and $d$ be the segments of the tangents to the sphere from the vertices of the tetrahedron. Is it true that that some of these segments necessarily form a triangle? (It is not obligatory to use all segments. The side of the triangle can be formed by two segments)
24. A sphere is inscribed into a prism $A B C A^{\prime} B^{\prime} C^{\prime}$ and touches its lateral faces $B C C^{\prime} B^{\prime}$ $C A A^{\prime} C^{\prime}, A B B^{\prime} A^{\prime}$ at points $A_{0}, B_{0}, C_{0}$ respectively. It is known that $$\angle A_{0} B B^{\prime}=\angle B_{0} C C^{\prime}=\angle C_{0} A A^{\prime}.$$ a) Find all possible values of these angles.
    b) Prove that segments $A A_{0}$, $B B_{0}$, $C C_{0}$ concur.
    c) Prove that the projections of the incenter to $A^{\prime} B^{\prime}, B^{\prime} C^{\prime}, C^{\prime} A^{\prime}$ are the vertices of a regular triangle.

DOWNLOAD SOLUTIONS