[Solutions] Sharygin Geometry Mathematical Olympiad 2019 (Final Round)

Grade 8

  1. A trapezoid with bases $AB$ and $CD$ is inscribed into a circle centered at $O$. Let $AP$ and $AQ$ be the tangents from $A$ to the circumcircle of triangle $CDO$. Prove that the circumcircle of triangle $APQ$ passes through the midpoint of $AB$. 
  2. A point $M$ inside triangle $ABC$ is such that $AM=AB/2$ and $CM=BC/2$. Points $C_0$ and $A_0$ lying on $AB$ and $CB$ respectively are such that $BC_0:AC_0 = BA_0:CA_0 = 3$. Prove that the distances from $M$ to $C_0$ and $A_0$ are equal. 
  3. Construct a regular triangle using a plywood square. (You can draw a line through pairs of points lying on the distance less than the side of the square, construct a perpendicular from a point to the line the distance between them does not exceed the side of the square, and measure segments on the constructed lines equal to the side or to the diagonal of the square) 
  4. Let $O$, $H$ be the orthocenter and circumcenter of of an acute-angled triangke $ABC$ with $AB<AC$. Let $K$ be the midpoint of $AH$. The line through $K$ perpendicular to $OK$ meet $AB$ and the tangent to the circumcircle at $A$ at $X$ and $Y$ respectively. Prove that $\angle XOY=\angle AOB$ 
  5. A point $H$ lies on the side $AB$ of regular polygon $ABCDE$. A circle with center $H$ and radius $HE$ meets the segments $DE$ and $CD$ at points $G$ and $F$ respectively. It is known that $DG=AH$. Prove that $CF=AH$. 
  6. Let points $M$ and $N$ lie on sides $AB$ and $BC$ of triangle $ABC$ in such a way that $MN||AC$. Points $M'$ and $N'$ are the reflections of $M$ and $N$ about $BC$ and $AB$ respectively. Let $M'A$ meet $BC$ at $X$, and let $N'C$ meet $AB$ at $Y$. Prove that $A$, $C$, $X$, $Y$ are concyclic. 
  7. What is the least positive integer $k$ such that, in every convex 1001-gon, the sum of any k diagonals is greater than or equal to the sum of the remaining diagonals? 

Grade 9

  1. 1 A triangle $OAB$ with $\angle A=90^{\circ}$ lies inside another triangle with vertex $O$. The altitude of $OAB$ from $A$ until it meets the side of angle $O$ at $M$. The distances from $M$ and $B$ to the second side of angle $O$ are $2$ and $1$ respectively. Find the length of $OA$. 
  2. Let $P$ be a point on the circumcircle of triangle $ABC$. Let $A_1$ be the reflection of the orthocenter of triangle $PBC$ about the reflection of the perpendicular bisector of $BC$. Points $B_1$ and $C_1$ are defined similarly. Prove that $A_1,B_1,C_1$ are collinear. 
  3. Let $ABCD$ be a cyclic quadrilateral such that $AD=BD=AC$. A point $P$ moves along the circumcircle $\omega$ of triangle $ABCD$. The lined $AP$ and $DP$ meet the lines $CD$ and $AB$ at points $E$ and $F$ respectively. The lines $BE$ and $CF$ meet point $Q$. Find the locus of $Q$. 
  4. A ship tries to land in the fog. The crew does not know the direction to the land. They see a lighthouse on a little island, and they understand that the distance to the lighthouse does not exceed $10 km$ (the exact distance is not known). The distance from the lighthouse to the land equals $10 km$. The lighthouse is surrounded by reefs, hence the ship cannot approach it. Can the ship land having sailed the distance not greater than $75 km$?. (The waterside is a straight line, the trajectory has to be given before the beginning of the motion, after that the autopilot navigates the ship.) 
  5. Let $R $ be the circumradius of a circumscribed quadrilateral $ABCD $. Let $h_1$ and $h_2$ be the altitudes from $A $ to $BC $ and $CD $ respectively. Similarly $h_3$ and $h_4$ are the altitudes from $C $ to $AB $ and $AD$. Prove that $$\frac {h_1+h_2- 2R}{h_1h_2}=\frac {h_3+h_4-2R}{h_3h_4}.$$
  6. A non-convex polygon has the property that every three consecutive its vertices from a right-angled triangle. Is it true that this polygon has always an angle equal to $90^{\circ} $ or to $270^{\circ} $? 
  7. Let the incircle $\omega $ of $\triangle ABC $ touch $AC $ and $AB $ at points $E $ and $F $ respectively. Points $X $, $Y $ of $\omega $ are such that $\angle BXC=\angle BYC=90^{\circ} $. Prove that $EF $ and $XY $ meet on the medial line of $ABC $. 
  8. A hexagon $A_1A_2A_3A_4A_5A_6$ has no four concyclic vertices, and its diagonals $A_1A_4$, $A_2A_5$ and $A_3A_6$ concur. Let $l_i $ be the radical axis of circles $A_iA_{i+1}A_{i-2} $ and $A_iA_{i-1}A_{i+2} $ (the points $A_i $ and $A_{i+6} $ coincide). Prove that $l_i, i=1,\cdots,6$, concur.

Grade 10

  1. Given a triangle $ABC$ with $\angle A = 45^\circ$. Let $A'$ be the antipode of $A$ in the circumcircle of $ABC$. Points $E$ and $F$ on segments $AB$ and $AC$ respectively are such that $A'B = BE$, $A'C = CF$. Let $K$ be the second intersection of circumcircles of triangles $AEF$ and $ABC$. Prove that $EF$ bisects $A'K$. 
  2. Let $A_1$, $B_1$, $C_1$ be the midpoints of sides $BC$, $AC$ and $AB$ of triangle $ABC$, $AK$ be the altitude from $A$, and $L$ be the tangency point of the incircle $\gamma$ with $BC$. Let the circumcircles of triangles $LKB_1$ and $A_1LC_1$ meet $B_1C_1$ for the second time at points $X$ and $Y$ respectively, and $\gamma$ meet this line at points $Z$ and $T$. Prove that $XZ = YT$. 
  3. Let $P$ and $Q$ be isogonal conjugates inside triangle $ABC$. Let $\omega$ be the circumcircle of $ABC$. Let $A_1$ be a point on arc $BC$ of $\omega$ satisfying $\angle BA_1P = \angle CA_1Q$. Points $B_1$ and $C_1$ are defined similarly. Prove that $AA_1$, $BB_1$, $CC_1$ are concurrent. 
  4. Prove that the sum of two nagelians is greater than the semiperimeter of a triangle. (The nagelian is the segment between the vertex of a triangle and the tangency point of the opposite side with the correspondent excircle.) 
  5. Let $AA_1$, $BB_1$, $CC_1$ be the altitudes of triangle $ABC$, and $A_0$, $C_0$ be the common points of the circumcircle of triangle $A_1BC_1$ with the lines $A_1B_1$ and $C_1B_1$ respectively. Prove that $AA_0$ and $CC_0$ meet on the median of ABC or are parallel to it.
  6. Let $AK$ and $AT$ be the bisector and the median of an acute-angled triangle $ABC$ with $AC > AB$. The line $AT$ meets the circumcircle of $ABC$ at point $D$. Point $F$ is the reflection of $K$ about $T$. If the angles of $ABC$ are known, find the value of angle $FDA$. 
  7. Let $P$ be an arbitrary point on side $BC$ of triangle $ABC$. Let $K$ be the incenter of triangle $PAB$. Let the incircle of triangle $PAC$ touch $BC$ at $F$. Point $G$ on $CK$ is such that $FG || PK$. Find the locus of $G$. 
  8. Several points and planes are given in the space. It is known that for any two of given points there exactly two planes containing them, and each given plane contains at least four of given points. Is it true that all given points are collinear?.

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