Sharygin Geometry Olympiad 2025-2026

Let $AK$ be the bisector of a triangle $ABC$, $N$ be a point on $AC$ such that $\angle NKC=\angle CAB/2$, and $L$ be the midpoint of $KN$. Prove that $\angle KBN=\angle LAK$.

Let $I$ be the incenter of a triangle $ABC$. The perpendicular bisector to $AI$ meets $BC$ at point $D$; the line $AD$ meets for the second time the circumcircle of $ABC$ at point $X$. Prove that $|BX-CX|=AX$.

Let $O$ be the circumcenter of a triangle $ABC$, $I$ be its incenter, $H$ be the orthocenter, and $N$ be the Nagel point. Prove that $IN=IH$ if and only if $ONH$ is a right angle.

The side $AB$ of a triangle $ABC$ touches the incircle and the excircle at points $P$ and $Q$ respectively. Let $T$ be the projection of the midpoint of $AB$ to the bisector of angle $C$. Prove that $C$, $P$, $Q$, $T$ are concyclic.

Let $ABC$ be a triangle with $\angle B=30^{\circ}$, $O$ be the circumcenter of $ABC$, $I$ be its incenter. The circles $AIB$, $CIB$ meet $BC$, $AB$ respectively at points $D$, $E$. Prove that $D$ is the orthocenter of triangle $OEI$.

Let $ABCD$ be a circumscribed quadrilateral with incenter $I$. The circles $BID$ and $AIC$ meet at point $P$, and the rays $AB$ and $DC$ meet at point $Q$. Let $R$ be the midpoint of $PI$. Prove that the quadrilateral $ARQD$ is cyclic.

A circle $\omega$, a point $A$ on it, and a point $B$ are given. Let $X$ be an arbitrary point of $\omega$, and $T$ be the common point of tangents to the circle $ABX$ at $X$ and $B$. Find the locus of points $T$.

Let $P$ and $Q$ lie on the side $AC$ of a triangle $ABC$ in such a way that $PQ=AC/2$. The point $B'$ is the reflection of $B$ about $AC$. Let $D$ and $E$ be the points on $BP$ and $BQ$ such that the lines $AD$ and $CE$ touch the circles $APB'$ and $CQB'$ respectively. Prove that the circumcircle of triangle $BDE$ touches $AC$.

The vertices of a right-angled triangle $ABC$ are points with integer coordinates. Its incircle centered at $I$ touches $AB$, $BC$ at points $C'$, $A'$ respectively. The lines $AA'$ and $CC'$ meet at point $G$. Prove that the line $IG$ passes through some point with integer coordinates.

Let $A_1\ldots A_n$ be a convex polygon. The points $A_1,\ldots, A_n$ in some order are vertices of two closed broken lines. What is the maximal possible ratio of their lengths?

A triangle $ABC$ ($AB<AC$) is given. Let $P$ be a point on the ray $BA$ such that $BP=AC$, and $Q$ be a point on the ray $CA$ such that $CQ=AB$. Let $BB_1$ and $CC_1$ be the perpendiculars to the line $PQ$. Prove that the circles $(CB_1Q)$, $(BC_1P)$ and the external bisector of angle $BAC$ have a common point.

Prove that the Nagel point of a triangle lies on its incircle if and only if the bisectors of two angles meet the side of the Gergonne triangle cutting the third vertex, at two points such that the segment between them equals a half of this side.

The line passing through the common point of the diagonals of a trapezoid $ABCD$ and parallel to its bases meets the lateral side $AB$ at point $M$. Let $K$ be the projection of $M$ to $CD$. Prove that $KM$ bisects the angle $AKB$.

A point $P$ inside a triangle $ABC$ is given. The lines $BP, CP$ meet the circle $ABC$ for the second time at points $E, F$ respectively. The circle $\Omega$ passes through $P, E$ and meets $AC$ at points $B_1, B_2$. The lines $PB_1,PB_2$ meet $AB$ at points $C_1, C_2$. Prove that $C_1, C_2, P, F$ are concyclic.

The incircle of a triangle $ABC$ centered at $I$ touches $BC$, $CA$, $AB$ at points $A'$, $B'$, $C'$ respectively. Let $A_b$, $A_c$, $B_a$, $C_a$ be the midpoints of segments $A'B$, $A'C$, $B'A$, $C'A$ respectively. The lines $A_bB_a$ and $A_cC_a$ meet at point $P$. Prove that the reflection of $I$ about $P$ lies on $AA'$.

The altitudes $AA_1$, $BB_1$, $CC_1$ of a triangle $ABC$ meet its circumcircle for the second time at points $A_2$, $B_2$, $C_2$ respectively. Let $A_3$ be the common point of circles $ABC$ and $AB_1C_1$, distinct from $A$; the points $B_3$, $C_3$ are defined similarly; $A_4$, $B_4$, $C_4$ are the feet of altitudes of triangle $A_1B_1C_1$. Prove that the lines $AA_4$, $BB_4$, $CC_4$, $A_2A_3$, $B_2B_3$, $C_2C_3$ concur.

Let $ABC$ be a triangle with $\angle A=2\pi/3$; $P$ be an arbitrary point inside this triangle lying on the bisector of angle $A$; the lines $BP$, $CP$ meet $AC$, $AB$ at points $E$, $F$ respectively; $D$ be an arbitrary point on the side $BC$; the lines $DE$, $DF$ meet $PC$, $PB$ at points $M$, $N$ respectivly. Find the value of angle $MAN$.

The incircle of a triangle $ABC$ touches $BC$ at point $D$. Let $F$ be the Feuerbach point, $H$ be the projection of $A$ to $DF$. Prove that $FH:DF=1:2$.