Sharygin Geometry Olympiad 2024

Cheenta hosted the final round of prestigious Sharygin Geometry Olympiad in India. The olympiad is intended for high-school students of four eldest grades. This post contains the problems from this contest.

Sharygin Olympiad is conducted by organisers from esteemed institutions in Russia.

1. Steklov Mathematical Institute RAS
2. Department of Education, Moscow City
3. Moscow Center for Pedagogical Mastery
4. Moscow Center for Continuous Mathematical Education
5. Journal of Classical Geometry
6. Moscow Institute of Physics and Technology

Awardees from India - 2024

1. ⁠Rishav Dutta (3rd prize)
2. ⁠⁠Krishiv Khandelwal (3rd Prize)
3. Sayantan Mazumdar (3rd Prize)
4. ⁠⁠Debarchan Neogi (3rd Prize)
5. Aratrik Pal (2nd Prize)
6. ⁠⁠Kanav Talwar (1st Prize)

First Day Problems - Grade 8

• A circle $\boldsymbol{\omega}$ centered at $O$ and a point $P$ inside it are given. Let $X$ be an arbitrary point of $\omega$, the line $X P$ and the circle $X O P$ meet $\omega$ for a second time at points $X_1, X_2$ respectively. Prove that all lines $X_1 X_2$ are parallel.
• Let $C M$ be the median of an acute-angled triangle $A B C$, and $P$ be the projection of the orthocenter $H$ to the bisector of angle $C$. Prove that $M P$ bisects the segment $C H$.
• Let $A D$ be the altitude of an acute-angled triangle $A B C$, and $A^{\prime}$ be the point of its circumcircle opposite to $A$. A point $P$ lies on the segment $A D$, and points $X, Y$ lie on the segments $A B, A C$ respectively in such a way that $\angle C B P=\angle A D Y$, $\angle B C P=\angle A D X$. Let $P A^{\prime}$ meet $B C$ at point $T$. Prove that $D, X, Y, T$ are concyclic.
• A square with sidelength 1 is cut from the paper. Construct a segment with length $1 / 2024$ using at most 20 folds. No instruments are available, it is allowed only to fold the paper and to mark the common points of folding lines.

First Day Problems - Grade 9

• Let $H$ be the orthocenter of an acute-angled triangle $A B C ; A_1, B_1, C_1$ be the touching points of the incircle with $B C, C A, A B$ respectively; $E_A, E_B$, $E_C$ be the midpoints of $\mathrm{AH}, \mathrm{BH}, \mathrm{CH}$ respectively. The circle centered at $E_A$ and passing through $A$ meets for the second time the bisector of angle $A$ at $A_2$; points $B_2, C_2$ are defined similarly. Prove that the triangles $A_1 B_1 C_1$ and $A_2 B_2 C_2$ are similar.
• Points $A, B, C, D$ on the plane do not form a rectangle. Let the sidelengths of triangle $T$ equal $A B+C D, A C+B D, A D+B C$. Prove that the triangle $T$ is acute-angled.
• Let $\left(P, P^{\prime}\right)$ and $\left(Q, Q^{\prime}\right)$ be two pairs of points isogonally conjugated with respect to a triangle $A B C$, and $R$ be the common point of lines $P Q$ and $P^{\prime} Q^{\prime}$. Prove that the pedal circles of points $P, Q$, and $R$ are coaxial.
• For which $n>0$ it is possible to mark several different points and several different circles on the plane in such a way that:
  • exactly $n$ marked circles pass through each marked point;
  • exactly $n$ marked points lie on each marked circle;
  • the center of each marked circle is marked?

First Day Problems - Grade 10

• The diagonals of a cyclic quadrilateral $A B C D$ meet at point $P$. The bisector of angle $A B D$ meets $A C$ at point $E$, and the bisector of angle $A C D$ meets $B D$ at point $F$. Prove that the lines $A F$ and $D E$ meet on the median of triangle $A P D$.
• For which greatest $n$ there exists a convex polyhedron with $n$ faces having the following property: for each face there exists a point outside the polyhedron such that the remaining $n-1$ faces are seen from this point?
• Let $B E$ and $C F$ be the bisectors of a triangle $A B C$. Prove that $2 E F \leq B F+C E$.
• Let $I$ be the incenter of a triangle $A B C$. The lines passing through $A$ and parallel to $B I, C I$ meet the perpendicular bisector to $A I$ at points $S$, $T$ respectively. Let $Y$ be the common point of $B T$ and $C S$, and $A^*$ be a point such that $BICA^*$ is a parallelogram. Prove that the midpoint of segment $Y A^*$ lies on the excircle of the triangle touching the side $B C$.

Second Day Problems - Grade 8

• The vertices $M, N, K$ of rectangle $K L M N$ lie on the sides $A B, B C, C A$ respectively of a regular triangle $A B C$ in such a way that $A M=2, K C=1$, the vertex $L$ lies outside the triangle. Find the value of angle $K M N$.
• A circle $\boldsymbol{\omega}$ touches lines $a$ and $b$ at points $A$ and $B$ respectively. An arbitrary tangent to the circle meets $a$ and $b$ at $X$ and $Y$ respectively. Points $X^{\prime}$ and $Y^{\prime}$ are the reflections of $X$ and $Y$ about $A$ and $B$ respectively. Find the locus of projections of the center of the circle to the lines $X^{\prime} Y^{\prime}$.
• A convex quadrilateral $A B C D$ is given. A line $l | A C$ meets the lines $A D, B C, A B, C D$ at points $X, Y, Z, T$ respectively. The circumcircles of triangles $X Y B$ and $Z T B$ meet for the second time at point $R$. Prove that $R$ lies on $B D$.
• Two polygons are cut from the cartboard. Is it possible that for any disposition of these polygons on the plane they have common inner point or have only finite number of common points?

Second Day Problems - Grade 9

• Let $A B C$ be an isosceles triangle $(A C=B C)$, $O$ be its circumcenter, $H$ be the orthocenter, and $P$ be a point inside the triangle such that $\angle A P H=$ $=\angle B P O=\pi / 2$. Prove that $\angle P A C=\angle P B A=\angle P C B$.
• The incircle of a triangle $A B C$ centered at $I$ touches the sides $B C, C A$, and $A B$ at points $A_1$, $B_1$, and $C_1$ respectively. The excircle centered at $J$ touches the side $A C$ at point $B_2$ and touches the extensions of $A B, B C$ at points $C_2, A_2$ respectively. Let the lines $I B_2$ and $J B_1$ meet at point $X$, the lines $I C_2$ and $J C_1$ meet at point $Y$, the lines $I A_2$ and $J A_1$ meet at point $Z$. Prove that if one of points $X, Y$, $Z$ lies on the incircle then two remaining points also lie on it.
• Let $P$ and $Q$ be arbitrary points on the side $B C$ of triangle $A B C$ such that $B P=C Q$. The common points of segments $A P$ and $A Q$ with the incircle form a quadrilateral $X Y Z T$. Find the locus of common points of diagonals of such quadrilaterals.
• Let points $P$ and $Q$ be isogonally conjugated with respect to a triangle $A B C$. The line $P Q$ meets the circumcircle of $A B C$ at point $X$. The reflection of $B C$ about $P Q$ meets $A X$ at point $E$. Prove that $A$, $P, Q, E$ are concyclic.

Second Day Problems - Grade 10

• The incircle of a right-angled triangle $A B C$ touches the hypothenuse $A B$ at point $T$. The squares $A T M P$ and $B T N Q$ lie outside the triangle. Prove that the areas of triangles $A B C$ and $T P Q$ are equal.
• A point $P$ lies on one of medians of triangle $A B C$ in such a way that $\angle P A B=\angle P B C=\angle P C A$. Prove that there exists a point $Q$ on another median such that $\angle Q B A=\angle Q C B=\angle Q A C$.
• Let $A B C$ be a triangle with $\angle A=60^{\circ} ; A D$, $B E$, and $C F$ be its bisectors; $P, Q$ be the projections of $A$ to $E F$ and $B C$ respectively; and $R$ be the second common point of the circle $D E F$ with $A D$. Prove that $P, Q, R$ are collinear.
• The common tangents to the circumcircle and an excircle of triangle $A B C$ meet $B C, C A, A B$ at points $A_1, B_1, C_1$ and $A_2, B_2, C_2$ respectively. The triangle $\Delta_1$ is formed by the lines $A A_1, B B_1$, and $C C_1$, the triangle $\Delta_2$ is formed by the lines $A A_2$, $B B_2$, and $C C_2$. Prove that the circumradii of these triangles are equal.