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[Solutions] Sharygin Geometry Mathematical Olympiad 2008 (Final Round)

Grade 8

  1. Does a convex quadrilateral without parallel sidelines exist such that it can be divided into four congruent triangles?
  2. Given right triangle $ ABC$ with hypothenuse $ AC$ and $ \angle A = 50^{\circ}$. Points $ K$ and $ L$ on the cathetus $ BC$ are such that $ \angle KAC = \angle LAB = 10^{\circ}$. Determine the ratio $ CK/LB$.
  3. Two opposite angles of a convex quadrilateral with perpendicular diagonals are equal. Prove that a circle can be inscribed in this quadrilateral.
  4. Let $ CC_0$ be a median of triangle $ ABC$; the perpendicular bisectors to $ AC$ and $ BC$ intersect $ CC_0$ in points $ A'$, $ B'$; $ C_1$ is the meet of lines $ AA'$ and $ BB'$. Prove that $ \angle C_1CA = \angle C_0CB$.
  5. Given two triangles $ ABC$, $ A'B'C'$. Denote by $ \alpha$ the angle between the altitude and the median from vertex $ A$ of triangle $ ABC$. Angles $ \beta$, $ \gamma$, $ \alpha'$, $ \beta'$, $ \gamma'$ are defined similarly. It is known that $ \alpha = \alpha'$, $ \beta = \beta'$, $ \gamma = \gamma'$. Can we conclude that the triangles are similar?
  6. Consider the triangles such that all their vertices are vertices of a given regular 2008-gon. What triangles are more numerous among them: acute-angled or obtuse-angled?
  7. Given isosceles triangle $ ABC$ with base $ AC$ and $ \angle B = \alpha$. The arc $ AC$ constructed outside the triangle has angular measure equal to $ \beta$. Two lines passing through $ B$ divide the segment and the arc $ AC$ into three equal parts. Find the ratio $ \alpha / \beta$.
  8. A convex quadrilateral was drawn on the blackboard. Boris marked the centers of four excircles each touching one side of the quadrilateral and the extensions of two adjacent sides. After this, Alexey erased the quadrilateral. Can Boris define its perimeter?

Grade 9

  1. A convex polygon can be divided into 2008 congruent quadrilaterals. Is it true that this polygon has a center or an axis of symmetry?
  2. Given quadrilateral $ ABCD$. Find the locus of points such that their projections to the lines $ AB$, $ BC$, $ CD$, $ DA$ form a quadrilateral with perpendicular diagonals.
  3. Prove the inequality \[ \frac1{\sqrt {2\sin A}} + \frac1{\sqrt {2\sin B}} + \frac1{\sqrt {2\sin C}}\leq\sqrt {\frac {p}{r}}, \] where $ p$ and $ r$ are the semiperimeter and the inradius of triangle $ ABC$.
  4. Let $ CC_0$ be a median of triangle $ ABC$; the perpendicular bisectors to $ AC$ and $ BC$ intersect $ CC_0$ in points $ A_c$, $ B_c$; $ C_1$ is the common point of $ AA_c$ and $ BB_c$. Points $ A_1$, $ B_1$ are defined similarly. Prove that circle $ A_1B_1C_1$ passes through the circumcenter of triangle $ ABC$.
  5. Can the surface of a regular tetrahedron be glued over with equal regular hexagons?
  6. Construct the triangle, given its centroid and the feet of an altitude and a bisector from the same vertex.
  7. The circumradius of triangle $ ABC$ is equal to $ R$. Another circle with the same radius passes through the orthocenter $ H$ of this triangle and intersect its circumcirle in points $ X$, $ Y$. Point $ Z$ is the fourth vertex of parallelogram $ CXZY$. Find the circumradius of triangle $ ABZ$.
  8. Points $ P$, $ Q$ lie on the circumcircle $ \omega$ of triangle $ ABC$. The perpendicular bisector $ l$ to $ PQ$ intersects $ BC$, $ CA$, $ AB$ in points $ A'$, $ B'$, $ C'$. Let $ A"$, $ B"$, $ C"$ be the second common points of $ l$ with the circles $ A'PQ$, $ B'PQ$, $ C'PQ$. Prove that $ AA"$, $ BB"$, $ CC"$ concur.

Grade 10

  1. An inscribed and circumscribed $ n$-gon is divided by some line into two inscribed and circumscribed polygons with different numbers of sides. Find $ n$.
  2. Let triangle $ A_1B_1C_1$ be symmetric to $ ABC$ wrt the incenter of its medial triangle. Prove that the orthocenter of $ A_1B_1C_1$ coincides with the circumcenter of the triangle formed by the excenters of $ ABC$.
  3. Suppose $ X$ and $ Y$ are the common points of two circles $ \omega_1$ and $ \omega_2$. The third circle $ \omega$ is internally tangent to $ \omega_1$ and $ \omega_2$ in $ P$ and $ Q$ respectively. Segment $ XY$ intersects $ \omega$ in points $ M$ and $ N$. Rays $ PM$ and $ PN$ intersect $ \omega_1$ in points $ A$ and $ D$; rays $ QM$ and $ QN$ intersect $ \omega_2$ in points $ B$ and $ C$ respectively. Prove that $ AB = CD$.
  4. Given three points $ C_0$, $ C_1$, $ C_2$ on the line $ l$. Find the locus of incenters of triangles $ ABC$ such that points $ A$, $ B$ lie on $ l$ and the feet of the median, the bisector and the altitude from $ C$ coincide with $ C_0$, $ C_1$, $ C_2$.
  5. A section of a regular tetragonal pyramid is a regular pentagon. Find the ratio of its side to the side of the base of the pyramid.
  6. The product of two sides in a triangle is equal to $ 8Rr$, where $ R$ and $ r$ are the circumradius and the inradius of the triangle. Prove that the angle between these sides is less than $ 60^{\circ}$.
  7. Two arcs with equal angular measure are constructed on the medians $ AA'$ and $ BB'$ of triangle $ ABC$ towards vertex $ C$. Prove that the common chord of the respective circles passes through $ C$.
  8. Given a set of points inn the plane. It is known that among any three of its points there are two such that the distance between them doesn't exceed 1. Prove that this set can be divided into three parts such that the diameter of each part does not exceed 1.

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MOlympiad: [Solutions] Sharygin Geometry Mathematical Olympiad 2008 (Final Round)
[Solutions] Sharygin Geometry Mathematical Olympiad 2008 (Final Round)
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