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[Solutions] Sharygin Geometry Mathematical Olympiad 2016 (Final Round)

Grade 8

  1. An altitude $AH$ of triangle $ABC$ bisects a median $BM$. Prove that the medians of triangle $ABM$ are sidelengths of a right-angled triangle.
  2. A circumcircle of triangle $ABC$ meets the sides $AD$ and $CD$ of a parallelogram $ABCD$ at points $K$ and $L$ respectively. Let $M$ be the midpoint of arc $KL$ not containing $B$. Prove that $DM \perp AC$.
  3. A trapezoid $ABCD$ and a line $l$ perpendicular to its bases $AD$ and $BC$ are given. A point $X$ moves along $l$. The perpendiculars from $A$ to $BX$ and from $D$ to $CX$ meet at point $Y$. Find the locus of $Y$.
  4. Is it possible to dissect a regular decagon along some of its diagonals so that the resulting parts can form two regular polygons?
  5. Three points are marked on the transparent sheet of paper. Prove that the sheet can be folded along some line in such a way that these points form an equilateral triangle.
  6. A triangle $ABC$ with $\widehat A = 60^0$ is given. Points $M$ and $N$ on $AB$ and $AC$ respectively are such that the circumcenter of $ABC$ bisects segment $MN$. Find the ratio $AN : MB$.
  7. Diagonals of a quadrilateral $ABCD$ are equal and meet at point $O$. The perpendicular bisectors to segments $AB$ and $CD$ meet at point $P$, and the perpendicular bisectors to $BC$ and $AD$ meet at point $Q$. Find angle $POQ$.
  8. A criminal is at point $X$, and three policemen at points $A$, $B$ and $C$ block him up, i.e. the point $X$ lies inside the triangle $ABC$. Each evening one of the policemen is replaced in the following way: a new policeman takes the position equidistant from three former policemen, after this one of the former policemen goes away so that three remaining policemen block up the criminal too. May the policemen after some time occupy again the points $A$, $B$ and $C$ (it is known that at any moment $X$ does not lie on a side of the triangle)?

Grade 9

  1. The diagonals of a parallelogram $ABCD$ meet at point $O$. The tangent to the circumcircle of triangle $BOC$ at $O$ meets ray $CB$ at point $F$. The circumcircle of triangle $FOD$ meets $BC$ for the second time at point $G$. Prove that $AG=AB$.
  2. Let $H$ be the orthocenter of an acute-angled triangle $ABC$. Point $X_A$ lying on the tangent at $H$ to the circumcircle of triangle $BHC$ is such that $AH=AX_A$ and $X_A \not= H$. Points $X_B,X_C$ are defined similarly. Prove that the triangle $X_AX_BX_C$ and the orthotriangle of $ABC$ are similar.
  3. Let $O$ and $I$ be the circumcenter and incenter of triangle $ABC$. The perpendicular from $I$ to $OI$ meets $AB$ and the external bisector of angle $C$ at points $X$ and $Y$ respectively. In what ratio does $I$ divide the segment $XY$?
  4. One hundred and one beetles are crawling in the plane. Some of the beetles are friends. Every one hundred beetles can position themselves so that two of them are friends if and only if they are at unit distance from each other. Is it always true that all one hundred and one beetles can do the same?
  5. The center of a circle $\omega_2$ lies on a circle $\omega_1$. Tangents $XP$ and $XQ$ to $\omega_2$ from an arbitrary point $X$ of $\omega_1$ ($P$ and $Q$ are the touching points) meet $\omega_1$ for the second time at points $R$ and $S$. Prove that the line $PQ$ bisects the segment $RS$.
  6. The sidelines $AB$ and $CD$ of a trapezoid meet at point $P$, and the diagonals of this trapezoid meet at point $Q$. Point $M$ on the smallest base $BC$ is such that $AM=MD$. Prove that $\angle PMB=\angle QMB$.
  7. From the altitudes of an acute-angled triangle, a triangle can be composed. Prove that a triangle can be composed from the bisectors of this triangle.
  8. The diagonals of a cyclic quadrilateral meet at point $M$. A circle $\omega$ touches segments $MA$ and $MD$ at points $P,Q$ respectively and touches the circumcircle of $ABCD$ at point $X$. Prove that $X$ lies on the radical axis of circles $ACQ$ and $BDP$.

Grade 10

  1. A line parallel to the side $BC$ of a triangle $ABC$ meets the sides $AB$ and $AC$ at points $P$ and $Q$, respectively. A point $M$ is chosen inside the triangle $APQ$. The segments $MB$ and $MC$ meet the segment $PQ$ at points $E$ and $F$, respectively. Let $N$ be the second intersection point of the circumcircles of the triangles $PMF$ and $QME$. Prove that the points $A,M,N$ are collinear.
  2. Let $I$ and $I_a$ be the incenter and excenter (opposite vertex $A$) of a triangle $ABC$, respectively. Let $A'$ be the point on its circumcircle opposite to $A$, and $A_1$ be the foot of the altitude from $A$. Prove that $\angle IA_1I_a=\angle IA'I_a$.
  3. Assume that the two triangles $ABC$ and $A'B'C'$ have the common incircle and the common circumcircle. Let a point $P$ lie inside both the triangles. Prove that the sum of the distances from $P$ to the sidelines of triangle $ABC$ is equal to the sum of distances from $P$ to the sidelines of triangle $A'B'C'$.
  4. The Devil and the Man play a game. Initially, the Man pays some cash $s$ to the Devil. Then he lists some $97$ triples $\{i,j,k\}$ consisting of positive integers not exceeding $100$. After that, the Devil draws some convex polygon $A_1A_2...A_{100}$ with area $100$ and pays to the Man, the sum of areas of all triangles $A_iA_jA_k$. Determine the maximal value of $s$ which guarantees that the Man receives at least as much cash as he paid. 
  5. Does there exist a convex polyhedron having equal number of edges and diagonals? (A diagonal of a polyhedron is a segment through two vertices not lying on the same face)
  6. A triangle $ABC$ is given. The point $K$ is the base of the external bisector of angle $A$. The point $M$ is the midpoint of the arc $AC$ of the circumcircle. The point $N$ on the bisector of angle $C$ is such that $AN \parallel BM$. Prove that the points $M,N,K$ are collinear. 
  7. Restore a triangle by one of its vertices, the circumcenter and the Lemoine's point. (The Lemoine's point is the intersection point of the reflections of the medians in the correspondent angle bisectors)
  8. Let $ABC$ be a non-isosceles triangle, let $AA_1$ be its angle bisector and $A_2$ be the touching point of the incircle with side $BC$. The points $B_1$, $B_2$, $C_1$, $C_2$ are defined similarly. Let $O$ and $I$ be the circumcenter and the incenter of triangle $ABC$. Prove that the radical center of the circumcircle of the triangles $AA_1A_2$, $BB_1B_2$, $CC_1C_2$ lies on the line $OI$.

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MOlympiad: [Solutions] Sharygin Geometry Mathematical Olympiad 2016 (Final Round)
[Solutions] Sharygin Geometry Mathematical Olympiad 2016 (Final Round)
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