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[Solutions] Sharygin Geometry Mathematical Olympiad 2017 (Correspondence Round)

  1. Mark on a cellular paper four nodes forming a convex quadrilateral with the sidelengths equal to four different primes.
  2. A circle cuts off four right-angled triangles from rectangle $A B C D$. Let $A_{0}, B_{0}, C_{0}$ and $D_{0}$ be the midpoints of the correspondent hypotenuses. Prove that $A_{0} C_{0}=B_{0} D_{0}$
  3. Let $I$ be the incenter of triangle $A B C ; H_{B}, H_{C}$ the orthocenters of triangles $A C I$ and $A B I$ respectively; $K$ the touching point of the incircle with the side $B C .$ Prove that $H_{B}$ $H_{C}$ and $K$ are collinear.
  4. A triangle $A B C$ is given. Let $C^{\prime}$ be the vertex of an isosceles triangle $A B C^{\prime}$ with $\angle C^{\prime}=120^{\circ}$ constructed on the other side of $A B$ than $C,$ and $B^{\prime}$ be the vertex of an equilateral triangle $A C B^{\prime}$ constructed on the same side of $A C$ as $A B C$. Let $K$ be the midpoint of $B B^{\prime}$. Find the angles of triangle $K C C^{\prime}$.
  5. A segment $A B$ is fixed on the plane. Consider all acute-angled triangles with side $A B$. Find the locus of a) the vertices of their greatest angles; b) their incenters.
  6. Let $A B C D$ be a convex quadrilateral with $A C=B D=A D ; E$ and $F$ the midpoints of $A B$ and $C D$ respectively; $O$ the common point of the diagonals. Prove that $E F$ passes through the touching points of the incircle of triangle $A O D$ with $A O$ and $O D$.
  7. The circumcenter of a triangle lies on its incircle. Prove that the ratio of its greatest and smallest sides is less than two.
  8. Let $A D$ be the base of trapezoid $A B C D$. It is known that the circumcenter of triangle $A B C$ lies on $B D$. Prove that the circumcenter of triangle $A B D$ lies on $A C$.
  9. Let $C_{0}$ be the midpoint of hypotenuse $A B$ of triangle $A B C ; A A_{1}, B B_{1}$ the bisectors of this triangle; $I$ its incenter. Prove that the lines $C_{0} I$ and $A_{1} B_{1}$ meet on the altitude from $C$
  10. Points $K$ and $L$ on the sides $A B$ and $B C$ of parallelogram $A B C D$ are such that $\angle A K D=\angle C L D .$ Prove that the circumcenter of triangle $B K L$ is equidistant from $A$ and $C$.
  11. A finite number of points is marked on the plane. Each three of them are not collinear. A circle is circumscribed around each triangle with marked vertices. Is it possible that all centers of these circles are also marked?
  12. Let $A A_{1}, C C_{1}$ be the altitudes of triangle $A B C, B_{0}$ the common point of the altitude from $B$ and the circumcircle of $A B C ;$ and $Q$ the common point of the circumcircles of $A B C$ and $A_{1} C_{1} B_{0},$ distinct from $B_{0} .$ Prove that $B Q$ is the symmedian of $A B C$.
  13. Two circles pass through points $A$ and $B$. A third circle touches both these circles and meets $A B$ at points $C$ and $D .$ Prove that the tangents to this circle at these points are parallel to the common tangents of two given circles.
  14. Let points $B$ and $C$ lie on the circle with diameter $A D$ and center $O$ on the same side of $A D .$ The circumcircles of triangles $A B O$ and $C D O$ meet $B C$ at points $F$ and $E$ respectively. Prove that $R^{2}=A F \cdot D E,$ where $R$ is the radius of the given circle.
  15. Let $A B C$ be an acute-angled triangle with incircle $\omega$ and incenter $I$. Let $\omega$ touch $A B, B C$ and $C A$ at points $D, E, F$ respectively. The circles $\omega_{1}$ and $\omega_{2}$ centered at $J_{1}$ and $J_{2}$ respectively are inscribed into $A D I F$ and $B D I E .$ Let $J_{1} J_{2}$ intersect $A B$ at point $M .$ Prove that $C D$ is perpendicular to $I M .$
  16. The tangents to the circumcircle of triangle $A B C$ at $A$ and $B$ meet at point $D$. The circle passing through the projections of $D$ to $B C, C A, A B,$ meet $A B$ for the second time at point $C^{\prime} .$ Points $A^{\prime}, B^{\prime}$ are defined similarly. Prove that $A A^{\prime}, B B^{\prime}, C C^{\prime}$ concur.
  17. Using a compass and a ruler, construct a point $K$ inside an acute-angled triangle $A B C$ so that $\angle K B A=2 \angle K A B$ and $\angle K B C=2 \angle K C B$
  18. Let $L$ be the common point of the symmedians of triangle $A B C,$ and $B H$ be its altitude. It is known that $\angle A L H=180^{\circ}-2 \angle A$. Prove that $\angle C L H=180^{\circ}-2 \angle C$.
  19. Let cevians $A A^{\prime}, B B^{\prime}$ and $C C^{\prime}$ of triangle $A B C$ concur at point $P$. The circumcircle of triangle $P A^{\prime} B^{\prime}$ meets $A C$ and $B C$ at points $M$ and $N$ respectively, and the circumcircles of triangles $P C^{\prime} B^{\prime}$ and $P A^{\prime} C^{\prime}$ meet $A C$ and $B C$ for the second time respectively at points $K$ and $L .$ The line $c$ passes through the midpoints of segments $M N$ and $K L .$ The lines $a$ and $b$ are defined similarly. Prove that $a, b$ and $c$ concur.
  20. Given a right-angled triangle $A B C$ and two perpendicular lines $x$ and $y$ passing through the vertex $A$ of its right angle. For an arbitrary point $X$ on $x$ define $y_{B}$ and $y_{C}$ as the reflections of $y$ about $X B$ and $X C$ respectively. Let $Y$ be the common point of $y_{b}$ and $y_{c}$. Find the locus of $Y$ (when $y_{b}$ and $y_{c}$ do not coincide).
  21. A convex hexagon is circumscribed about a circle of radius 1. Consider the three segments joining the midpoints of its opposite sides. Find the greatest real number $r$ such that the length of at least one segment is at least $r .$
  22. Let $P$ be an arbitrary point on the diagonal $A C$ of cyclic quadrilateral $A B C D$, and $P K, P L, P M, P N, P O$ be the perpendiculars from $P$ to $A B$, $B C$, $C D$, $D A$, $B D$ respectively. Prove that the distance from $P$ to $K N$ is equal to the distance from $O$ to $M L$
  23. Let a line $m$ touch the incircle of triangle $A B C$. The lines passing through the incenter $I$ and perpendicular to $A I$, $B I$, $C I$ meet $m$ at points $A^{\prime}, B^{\prime}, C^{\prime}$ respectively. Prove that $A A^{\prime}$, $B B^{\prime}$ and $C C^{\prime}$ concur.
  24. Two tetrahedrons are given. Each two faces of the same tetrahedron are not similar, but each face of the first tetrahedron is similar to some face of the second one. Does this yield that these tetrahedrons are similar?

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MOlympiad: [Solutions] Sharygin Geometry Mathematical Olympiad 2017 (Correspondence Round)
[Solutions] Sharygin Geometry Mathematical Olympiad 2017 (Correspondence Round)
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https://www.molympiad.net/2018/08/sharygin-geometry-mathematical-olympiad-2017-correspondence.html
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https://www.molympiad.net/2018/08/sharygin-geometry-mathematical-olympiad-2017-correspondence.html
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