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

Grade 8

  1. Let $ABCD$ be a cyclic quadrilateral with $AB = BC$ and $AD = CD$. A point $M$ lies on the minor arc $CD$ of its circumcircle. The lines $BM$ and $CD$ meet at point $P$, the lines $AM$ and $BD$ meet at point $Q$. Prove that $PQ ∥ AC$.
  2. Let $H$ and $O$ be the orthocenter and the circumcenter of an acute-angled triangle $ABC$, respectively. The perpendicular bisector to segment $BH$ meets $AB$ and $BC$ at points $A_1$ and $C_1$, respectively. Prove that $OB$ bisects the angle $A_1OC_1$.
  3. Let $AD$, $BE$ and $CF$ be the medians of triangle $ABC$. The points $X$ and $Y$ are the reflections of $F$ about $AD$ and $BE$, respectively. Prove that the circumcircles of triangles $BEX$ and $ADY$ are concentric.
  4. Alex dissects a paper triangle into two triangles. Each minute after this he dissects one of obtained triangles into two triangles. After several time (at least one hour) it appeared that all obtained triangles are congruent. Find all initial triangles for which this is possible.
  5. A square $ABCD$ is given. Two circles are inscribed into angles $A$ and $B$, and the sum of their diameters is equal to the sidelength of the square. Prove that one of their common tangents passes through the midpoint of $AB$.
  6. A median of an acute-angled triangle dissects it into two triangles. Prove that each of them can be covered by a semidisc congruent to a half of the circumdisc of the initial triangle.
  7. Let $A_1A_2 . . . A_{13}$ and $B_1B_2 . . . B_{13}$ be two regular $13$-gons in the plane such that the points $B_1$ and $A_{13}$ coincide and lie on the segment $A_1B_{13}$, and both polygons lie in the same semiplane with respect to this segment. Prove that the lines $A_1A_9$, $B_{13}B_8$ and $A_8B_9$ are concurrent.
  8. Let $ABCD$ be a square, and let $P$ be a point on the minor arc $CD$ of its circumcircle. The lines $PA$, $PB$ meet the diagonals $BD$, $AC$ at points $K$, $L$ respectively. The points $M$, $N$ are the projections of $K$, $L$ respectively to $CD$, and $Q$ is the common point of lines $KN$ and $ML$. Prove that $PQ$ bisects the segment $AB$.

Grade 9

  1. Let $ABC$ be a regular triangle. The line passing through the midpoint of $AB$ and parallel to $AC$ meets the minor arc $AB$ of its circumcircle at point $K$. Prove that the ratio $AK : BK$ is equal to the ratio of the side and the diagonal of a regular pentagon.
  2. Let $I$ be the incenter of triangle $ABC$, $M$ be the midpoint of $AC$, and $W$ be the midpoint of arc $AB$ of its circumcircle not containing $C$. It is known that $\widehat{AIM} = 90^0$. Find the ratio $CI : IW$.
  3. The angles $B$ and $C$ of an acute-angled triangle $ABC$ are greater than $60^0$. Points $P$ and $Q$ are chosen on the sides $AB$ and $AC$, respectively, so that the points $A$, $P$, $Q$ are concyclic with the orthocenter $H$ of the triangle $ABC$. Point $K$ is the midpoint of $PQ$. Prove that $\widehat{BKC} > 90^0$.
  4. Points $M$ and $K$ are chosen on lateral sides $AB$ and $AC$, respectively, of an isosceles triangle $ABC$, and point $D$ is chosen on its base $BC$ so that $AMDK$ is a parallelogram. Let the lines $MK$ and $BC$ meet at point $L$, and let $X$, $Y$ be the intersection points of $AB$, $AC$ respectively with the perpendicular line from $D$ to $BC$. Prove that the circle with center $L$ and radius $LD$ and the circumcircle of triangle $AXY$ are tangent.
  5. Let $BH_b$, $CH_c$ be altitudes of a triangle $ABC$. The line $H_bH_c$ meets the circumcircle of $ABC$ at points $X$ and $Y$. Points $P$ and $Q$ are the reflections of $X$ and $Y$ about $AB$ and $AC$, respectively. Prove that $PQ ∥ BC$.
  6. Let $ABC$ be a right-angled triangle ($\widehat C = 90^0$) and $D$ be the midpoint of an altitude from $C$. The reflections of the line $AB$ about $AD$ and $BD$, respectively, meet at point $F$. Find the ratio $S_{ABF} : S_{ABC}$.
  7. Let $a$ and $b$ be parallel lines with $50$ distinct points marked on a and $50$ distinct points marked on $b$. Find the greatest possible number of acute-angled triangles all whose vertices are marked.
  8. Let $AK$ and $BL$ be the altitudes of an acute-angled triangle $ABC$, and let $\omega$ be the excircle of $ABC$ touching the side $AB$. The common internal tangents to circles $CKL$ and $\omega$ meet $AB$ at points $P$ and $Q$. Prove that $AP = BQ$.

Grade 10

  1. Let $A$ and $B$ be the common points of two circles, and $CD$ be their common tangent ($C$ and $D$ are the tangency points). Let $O_a$, $O_b$ be the circumcenters of triangles $CAD$, $CBD$ respectively. Prove that the midpoint of segment $O_aO_b$ lies on the line $AB$.
  2. Prove that the distance from any vertex of an acute-angled triangle to the corresponding excenter is less than the sum of two greatest sidelengths.
  3. Let $ABCD$ be a convex quadrilateral, and let $\omega_A$, $\omega_B$, $\omega_C$, $\omega_D$ be the circumcircles of triangles $BCD$, $ACD$, $ABD$, $ABC$, respectively. Denote by $XA$ the product of the power of $A$ with respect to $\omega_A$ and the area of triangle $BCD$. Define $XB$, $XC$, $XD$ similarly. Prove that $$XA + XB + XC + XD = 0.$$
  4. A scalene triangle $ABC$ and its incircle ω are given. Using only a ruler and drawing at most eight lines, rays or segments, constructpoints $A′, B′, C′$ on $\omega$ such that the rays $B′C′$, $C′A′$, $A′B′$ pass through $A, B, C$, respectively.
  5. Let $BB′$, $CC′$ be the altitudes of an acute-angled triangle $ABC$. Two circles passing through $A$ and $C′$ are tangent to $BC$ at points $P$ and $Q$. Prove that $A, B′, P, Q$ are concyclic.
  6. Let the insphere of a pyramid $SABC$ touch the faces $SAB$, $SBC$, $SCA$ at points $D, E, F$ respectively. Find all possible values of the sum of angles $SDA$, $SEB$ and $SFC$.
  7. A quadrilateral $ABCD$ is circumscribed around circle $\omega$ centered at $I$ and inscribed into circle $\Gamma$. The lines $AB$ and $CD$ meet at point $P$, the lines $BC$ and $AD$ meet at point $Q$. Prove that the circles $PIQ$ and $\Gamma$ are orthogonal.
  8. Let $S$ be a set of points in the plane, $|S|$ is even; no three points of $S$ are collinear. Prove that $S$ can be partitioned into two sets $S_1$ and $S_2$ so that their convex hulls have equal number of vertices.

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MOlympiad: [Solutions] Sharygin Geometry Mathematical Olympiad 2017 (Final Round)
[Solutions] Sharygin Geometry Mathematical Olympiad 2017 (Final Round)
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https://www.molympiad.net/2017/09/sharygin-geometry-olympiad-2017-solutions.html
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