[Solutions] Sharygin Geometry Mathematical Olympiad 2013 (Correspondence Round)

1. Let $A B C$ be an isosceles triangle with $A B=B C .$ Point $E$ lies on the side $A B,$ and $E D$ is the perpendicular from $E$ to $B C .$ It is known that $A E=D E .$ Find $\angle D A C$.
2. Let $A B C$ be an isosceles triangle $(A C=B C)$ with $\angle C=20^{\circ} .$ The bisectors of angles $A$ and $B$ meet the opposite sides at points $A_{1}$ and $B_{1}$ respectively. Prove that the triangle $A_{1} O B_{1}($ where $O$ is the circumcenter of $A B C)$ is regular.
3. Let $A B C$ be a right-angled triangle $\left(\angle B=90^{\circ}\right)$. The excircle inscribed into the angle $A$ touches the extensions of the sides $A B$, $A C$ at points $A_{1}$, $A_{2}$ respectively; points $C_{1}$, $C_{2}$ are defined similarly. Prove that the perpendiculars from $A, B, C$ to $C_{1} C_{2}$, $A_{1} C_{1}$, $A_{1} A_{2}$ respectively, concur.
4. Let $A B C$ be a nonisosceles triangle. Point $O$ is its circumcenter, and point $K$ is the center of the circumcircle $w$ of triangle $B C O$. The altitude of $A B C$ from $A$ meets $w$ at a point $P .$ The line $P K$ intersects the circumcircle of $A B C$ at points $E$ and $F .$ Prove that one of the segments $E P$ and $F P$ is equal to the segment $P A$.
5. Four segments drawn from a given point inside a convex quadrilateral to its vertices, split the quadrilateral into four equal triangles. Can we assert that this quadrilateral is a rhombus?
6. Diagonals $A C$ and $B D$ of a trapezoid $A B C D$ meet at point $P .$ The circumcircles of triangles $A B P$ and $C D P$ intersect the line $A D$ for the second time at points $X$ and $Y$ respectively. Let $M$ be the midpoint of segment $X Y .$ Prove that $B M=C M$.
7. Let $B D$ be a bisector of triangle $A B C .$ Points $I_{a}, I_{c}$ are the incenters of triangles $A B D, C B D$ respectively. The line $I_{a} I_{c}$ meets $A C$ in point $Q .$ Prove that $\angle D B Q=90^{\circ} .$
8. Let $X$ be an arbitrary point inside the circumcircle of a triangle $A B C .$ The lines $B X$ and $C X$ meet that circumcircle in points $K$ and $L$ respectively. The line $L K$ intersects $B A$ and $A C$ at points $E$ and $F$ respectively. Find the locus of points $X$ such that the circumcircles of triangles $A F K$ and $A E L$ touch.
9. Let $T_{1}$ and $T_{2}$ be the points of tangency of the excircles of a triangle $A B C$ with its sides $B C$ and $A C$ respectively. It is known that the reflection of the incenter of $A B C$ across the midpoint of $A B$ lies on the circumcircle of triangle $C T_{1} T_{2} .$ Find $\angle B C A$.
10. The incircle of triangle $A B C$ touches the side $A B$ at point $C^{\prime} ;$ the incircle of triangle $A C C^{\prime}$ touches the sides $A B$ and $A C$ at points $C_{1}, B_{1} ;$ the incircle of triangle $B C C^{\prime}$ touches the sides $A B$ and $B C$ at points $C_{2}, A_{2} .$ Prove that the lines $B_{1} C_{1}, A_{2} C_{2},$ and $C C^{\prime}$ concur.
11. a) Let $A B C D$ be a convex quadrilateral and $r_{1} \leq r_{2} \leq r_{3} \leq r_{4}$ be the radii of the incircles of triagles $A B C$, $B C D$, $C D A$, $D A B$. Can the inequality $r_{4}>2 r_{3}$ hold?.
    b) The diagonals of a convex quadrilateral $A B C D$ meet in point $E$. Let $r_{1} \leq r_{2} \leq r_{3} \leq r_{4}$ be the radii of the incircles of triangles $A B E$, $B C E$, $C D E$, $D A E$. Can the inequality $r_{2}>2 r_{1}$ hold?
12. On each side of a triangle $A B C,$ two distinct points are marked. It is known that these points are the feet of the altitudes and of the bisectors.
    a) Using only a ruler determine which points are the feet of the altitudes and which points are the feet of the bisectors.
    b) Solve a) drawing only three lines.
13. Let $A_{1}$ and $C_{1}$ be the tangency points of the incircle of triangle $A B C$ with $B C$ and $A B$ respectively, $A^{\prime}$ and $C^{\prime}$ be the tangency points of the excircle inscribed into the angle $B$ with the extensions of $B C$ and $A B$ respectively. Prove that the orthocenter $H$ of triangle $A B C$ lies on $A_{1} C_{1}$ if and only if the lines $A^{\prime} C_{1}$ and $B A$ are orthogonal.
14. Let $M, N$ be the midpoints of diagonals $A C, B D$ of a right-angled trapezoid $A B C D$ $\left(\angle A=\angle D=90^{\circ}\right) .$ The circumcircles of triangles $A B N$, $C D M$ meet the line $B C$ in points $Q$, $R .$ Prove that the distances from $Q$, $R$ to the midpoint of $M N$ are equal.
15. a) Triangles $A_{1} B_{1} C_{1}$ and $A_{2} B_{2} C_{2}$ are inscribed into triangle $A B C$ so that $C_{1} A_{1} \perp$ $B C, A_{1} B_{1} \perp C A, B_{1} C_{1} \perp A B, B_{2} A_{2} \perp B C, C_{2} B_{2} \perp C A, A_{2} C_{2} \perp A B .$ Prove that these triangles are equal.
    b) Points $A_{1}$, $B_{1}$, $C_{1}$, $A_{2}$, $B_{2}$, $C_{2}$ lie inside a triangle $A B C$ so that $A_{1}$ is on segment $A B_{1}$, $B_{1}$ is on segment $B C_{1}$, $C_{1}$ is on segment $C A_{1}$, $A_{2}$ is on segment $A C_{2}$, $B_{2}$ is on segment $B A_{2}$, $C_{2}$ is on segment $C B_{2},$ and the angles $B A A_{1}$, $C B B_{1}$, $A C C_{1}$, $C A A_{2}$, $A B B_{2}$, $B C C_{2}$ are equal. Prove that the triangles $A_{1} B_{1} C_{1}$ and $A_{2} B_{2} C_{2}$ are equal.
16. The incircle of triangle $A B C$ touches $B C, C A, A B$ at points $A^{\prime}, B^{\prime}, C^{\prime}$ respectively. The perpendicular from the incenter $I$ to the median from vertex $C$ meets the line $A^{\prime} B^{\prime}$ in point $K .$ Prove that $C K \| A B$.
17. An acute angle between the diagonals of a cyclic quadrilateral is equal to $\phi$. Prove that an acute angle between the diagonals of any other quadrilateral having the same sidelengths is smaller than $\phi$.
18. Let $A D$ be a bisector of triangle $A B C .$ Points $M$ and $N$ are the projections of $B$ and $C$ respectively to $A D .$ The circle with diameter $M N$ intersects $B C$ at points $X$ and $Y .$ Prove that $\angle B A X=\angle C A Y .$
19. a) The incircle of a triangle $A B C$ touches $A C$ and $A B$ at points $B_{0}$ and $C_{0}$ respectively. The bisectors of angles $B$ and $C$ meet the perpendicular bisector to the bisector $A L$ in points $Q$ and $P$ respectively. Prove that the lines $P C_{0}, Q B_{0},$ and $B C$ concur.
    b) Let $A L$ be the bisector of a triangle $A B C .$ Points $O_{1}$ and $O_{2}$ are the circumcenters of triangles $A B L$ and $A C L$ respectively. Points $B_{1}$ and $C_{1}$ are the projections of $C$ and $B$ to the bisectors of angles $B$ and $C$ respectively. Prove that the lines $O_{1} C_{1}, O_{2} B_{1},$ and $B C$ concur. c) Prove that two points obtained in pp. a ) and b) coincide.
20. Let $C_{1}$ be an arbitrary point on the side $A B$ of triangle $A B C .$ Points $A_{1}$ and $B_{1}$ on the rays $B C$ and $A C$ are such that $\angle A C_{1} B_{1}=\angle B C_{1} A_{1}=\angle A C B$. The lines $A A_{1}$ and $B B_{1}$ meet in point $C_{2} .$ Prove that all the lines $C_{1} C_{2}$ have a common point.
21. Let $A$ be a point inside a circle $\omega$. One of two lines drawn through $A$ intersects $\omega$ at points $B$ and $C,$ the second one intersects it at points $D$ and $E(D$ lies between $A$ and $E) .$ The line passing through $D$ and parallel to $B C$ meets $\omega$ for the second time at point $F,$ and the line $A F$ meets $\omega$ at point $T .$ Let $M$ be the common point of the lines $E T$ and $B C,$ and $N$ be the reflection of $A$ across $M .$ Prove that the circumcircle of triangle $D E N$ passes through the midpoint of segment $B C$.
22. The common perpendiculars to the opposite sidelines of a nonplanar quadrilateral are mutually orthogonal. Prove that they intersect.
23. Two convex polytopes $A$ and $B$ do not intersect. The polytope $A$ has exactly 2012 planes of symmetry. What is the maximal number of symmetry planes of the union of $A$ and $B,$ if $B$ has
    a) $2012,$
    b) $2013$ symmetry planes?
    c) What is the answer to the question of b), if the symmetry planes are replaced by the symmetry axes?