[Solutions] Sharygin Geometry Mathematical Olympiad 2015 (Correspondence Round)

1. Tanya cut out a convex polygon from the paper, fold it several times and obtained a two-layers quadrilateral. Can the cutted polygon be a heptagon?
2. Let $O$ and $H$ be the circumcenter and the orthocenter of a triangle $A B C .$ The line passing through the midpoint of $O H$ and parallel to $B C$ meets $A B$ and $A C$ at points $D$ and $E .$ It is known that $O$ is the incenter of triangle $A D E .$ Find the angles of $A B C$.
3. The side $A D$ of a square $A B C D$ is the base of an obtuse-angled isosceles triangle $A E D$ with vertex $E$ lying inside the square. Let $A F$ be a diameter of the circumcircle of this triangle, and $G$ be a point on $C D$ such that $C G=D F$. Prove that angle $B G E$ is less than half of angle $A E D$.
4. In a parallelogram $A B C D$ the trisectors of angles $A$ and $B$ are drawn. Let $O$ be the common points of the trisectors nearest to $A B .$ Let $A O$ meet the second trisector of angle $B$ at point $A_{1},$ and let $B O$ meet the second trisector of angle $A$ at point $B_{1} .$ Let $M$ be the midpoint of $A_{1} B_{1} .$ Line $M O$ meets $A B$ at point $N .$ Prove that triangle $A_{1} B_{1} N$ is equilateral.
5. Let a triangle $A B C$ be given. Two circles passing through $A$ touch $B C$ at points $B$ and $C$ respectively. Let $D$ be the second common point of these circles $(A$ is closer to $B C$ than $D$ ). It is known that $B C=2 B D$. Prove that $\angle D A B=2 \angle A D B$.
6. Let $A A^{\prime}, B B^{\prime}$ and $C C^{\prime}$ be the altitudes of an acute-angled triangle $A B C$. Points $C_{a}, C_{b}$ are symmetric to $C^{\prime}$ wrt $A A^{\prime}$ and $B B^{\prime} .$ Points $A_{b}, A_{c}, B_{c}, B_{a}$ are defined similarly. Prove that lines $A_{b} B_{a}, B_{c} C_{b}$ and $C_{a} A_{c}$ are parallel.
7. The altitudes $A A_{1}$ and $C C_{1}$ of a triangle $A B C$ meet at point $H .$ Point $H_{A}$ is symmetric to $H$ about $A$. Line $H_{A} C_{1}$ meets $B C$ at point $C^{\prime} ;$ point $A^{\prime}$ is defined similarly. Prove that $A^{\prime} C^{\prime} \| A C$.
8. Diagonals of an isosceles trapezoid $A B C D$ with bases $B C$ and $A D$ are perpendicular. Let $D E$ be the perpendicular from $D$ to $A B,$ and let $C F$ be the perpendicular from $C$ to $D E .$ Prove that angle $D B F$ is equal to half of angle $F C D .$
9. Let $A B C$ be an acute-angled triangle. Construct points $A^{\prime}$, $B^{\prime},$ $C^{\prime}$ on its sides $B C$, $C A$, $A B$ such that $A^{\prime} B^{\prime} \| A B$, $C^{\prime} C$ is the bisector of angle $A^{\prime} C^{\prime} B^{\prime}$, $A^{\prime} C^{\prime}+B^{\prime} C^{\prime}=A B$.
10. The diagonals of a convex quadrilateral divide it into four similar triangles. Prove that is possible to inscribe a circle into this quadrilateral.
11. Let $H$ be the orthocenter of an acute-angled triangle $A B C$. The perpendicular bisector to segment $B H$ meets $B A$ and $B C$ at points $A_{0}, C_{0}$ respectively. Prove that the perimeter of triangle $A_{0} O C_{0}(O$ is the circumcenter of $\triangle A B C)$ is equal to $A C$.
12. Find the maximal number of discs which can be disposed on the plane so that each two of them have a common point and no three have it.
13. Let $A H_{1}, B H_{2}$ and $C H_{3}$ be the altitudes of a triangle $A B C .$ Point $M$ is the midpoint of $H_{2} H_{3} .$ Line $A M$ meets $H_{2} H_{1}$ at point $K .$ Prove that $K$ lies on the medial line of $A B C$ parallel to $A C$.
14. Let $A B C$ be an acute-angled, nonisosceles triangle. Point $A_{1}, A_{2}$ are symmetric to the feet of the internal and the external bisectors of angle $A$ wrt the midpoint of $B C$. Segment $A_{1} A_{2}$ is a diameter of a circle $\alpha .$ Circles $\beta$ and $\gamma$ are defined similarly. Prove that these three circles have two common points.
15. The sidelengths of a triangle $A B C$ are not greater than $1 .$ Prove that $p(1-2 R r)$ is not greater than 1 , where $p$ is the semiperimeter, $R$ and $r$ are the circumradius and the inradius of $A B C$.
16. The diagonals of a convex quadrilateral divide it into four triangles. Restore the quadrilateral by the circumcenters of two adjacent triangles and the incenters of two mutually opposite triangles.
17. Let $O$ be the circumcenter of a triangle $A B C .$ The projections of points $D$ and $X$ to the sidelines of the triangle lie on lines $l$ and $L$ such that $l \| X O .$ Prove that the angles formed by $L$ and by the diagonals of quadrilateral $A B C D$ are equal.
18. Let $A B C D E F$ be a cyclic hexagon, points $K, L, M, N$ be the common points of lines $A B$ and $C D, A C$ and $B D, A F$ and $D E, A E$ and $D F$ respectively. Prove that if three of these points are collinear then the fourth point lies on the same line.
19. Let $L$ and $K$ be the feet of the internal and the external bisector of angle $A$ of a triangle $A B C .$ Let $P$ be the common point of the tangents to the circumcircle of the triangle at $B$ and $C .$ The perpendicular from $L$ to $B C$ meets $A P$ at point $Q .$ Prove that $Q$ lies on the medial line of triangle $L K P$.
20. Given are a circle and an ellipse lying inside it with focus $C .$ Find the locus of the circumcenters of triangles $A B C,$ where $A B$ is a chord of the circle touching the ellipse.
21. A quadrilateral $A B C D$ is inscribed into a circle $\omega$ with center $O .$ Let $M_{1}$ and $M_{2}$ be the midpoints of segments $A B$ and $C D$ respectively. Let $\Omega$ be the circumcircle of triangle $O M_{1} M_{2} .$ Let $X_{1}$ and $X_{2}$ be the common points of $\omega$ and $\Omega,$ and $Y_{1}$ and $Y_{2}$ the second common points of $\Omega$ with the circumcircles of triangles $C D M_{1}$ and $A B M_{2}$. Prove that $X_{1} X_{2} \| Y_{1} Y_{2}$
22. The faces of an icosahedron are painted into 5 colors in such a way that two faces painted into the same color have no common points, even a vertices. Prove that for any point lying inside the icosahedron the sums of the distances from this point to the red faces and the blue faces are equal.
23. A tetrahedron $A B C D$ is given. The incircles of triangles $A B C$ and $A B D$ with centers $O_{1}, O_{2},$ touch $A B$ at points $T_{1}, T_{2} .$ The plane $\pi_{A B}$ passing through the midpoint of $T_{1} T_{2}$ is perpendicular to $O_{1} O_{2} .$ The planes $\pi_{A C}, \pi_{B C}, \pi_{A D}, \pi_{B D}, \pi_{C D}$ are defined similarly. Prove that these six planes have a common point.
24. The insphere of a tetrahedron $A B C D$ with center $O$ touches its faces at points $A_{1}, B_{1}, C_{1}$ и $D_{1}$.
    a) Let $P_{a}$ be a point such that its reflections in lines $O B, O C$ and $O D$ lie on plane $B C D$. Points $P_{b}, P_{c}$ and $P_{d}$ are defined similarly. Prove that lines $A_{1} P_{a}, B_{1} P_{b}, C_{1} P_{c}$ and $D_{1} P_{d}$ concur at some point $P$.
    b) Let $I$ be the incenter of $A_{1} B_{1} C_{1} D_{1}$ and $A_{2}$ the common point of line $A_{1} I$ with plane $B_{1} C_{1} D_{1} .$ Points $B_{2}, C_{2}, D_{2}$ are defined similarly. Prove that $P$ lies inside $A_{2} B_{2} C_{2} D_{2}$

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