This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 7

2022 Sharygin Geometry Olympiad, 8
Tags: Sharygin Geometry Olympiad , Sharygin 2022 , geometry
Points $P,Q,R$ lie on the sides $AB,BC,CA$ of triangle $ABC$ in such a way that $AP=PR, CQ=QR$. Let $H$ be the orthocenter of triangle $PQR$, and $O$ be the circumcenter of triangle $ABC$. Prove that $$OH||AC$$.

2022 Sharygin Geometry Olympiad, 7
Tags: Sharygin Geometry Olympiad , Sharygin 2022 , geometry
A square with center $F$ was constructed on the side $AC$ of triangle $ABC$ outside it. After this, everything was erased except $F$ and the midpoints $N,K$ of sides $BC,AB$. Restore the triangle.

2022 Sharygin Geometry Olympiad, 10
Tags: Sharygin Geometry Olympiad , Sharygin 2022 , geometry
Let $\omega_1$ be the circumcircle of triangle $ABC$ and $O$ be its circumcenter. A circle $\omega_2$ touches the sides $AB, AC$, and touches the arc $BC$ of $\omega_1$ at point $K$. Let $I$ be the incenter of $ABC$. Prove that the line $OI$ contains the symmedian of triangle $AIK$.

2022 Sharygin Geometry Olympiad, 6
Tags: Sharygin Geometry Olympiad , Sharygin 2022 , geometric inequality , inequalities
The incircle and the excircle of triangle $ABC$ touch the side $AC$ at points $P$ and $Q$ respectively. The lines $BP$ and $BQ$ meet the circumcircle of triangle $ABC$ for the second time at points $P'$ and $Q'$ respectively. Prove that $$PP' > QQ'$$

2022 Sharygin Geometry Olympiad, 15
Tags: geometry , Sharygin Geometry Olympiad , Sharygin 2022 , Inversion , similar triangles
A line $l$ parallel to the side $BC$ of triangle $ABC$ touches its incircle and meets its circumcircle at points $D$ and $E$. Let $I$ be the incenter of $ABC$. Prove that $AI^2 = AD \cdot AE$.

2022 Sharygin Geometry Olympiad, 11
Tags: Sharygin Geometry Olympiad , Sharygin 2022 , geometry
Let $ABC$ be a triangle with $\angle A=60^o$ and $T$ be a point such that $\angle ATB=\angle BTC=\angle ATC$. A circle passing through $B,C$ and $T$ meets $AB$ and $AC$ for the second time at points $K$ and $L$.Prove that the distances from $K$ and $L$ to $AT$ are equal.

2022 Sharygin Geometry Olympiad, 9
Tags: Sharygin Geometry Olympiad , Sharygin 2022 , geometry
The sides $AB, BC, CD$ and $DA$ of quadrilateral $ABCD$ touch a circle with center $I$ at points $K, L, M$ and $N$ respectively. Let $P$ be an arbitrary point of line $AI$. Let $PK$ meet $BI$ at point $Q, QL$ meet $CI$ at point $R$, and $RM$ meet $DI$ at point $S$. Prove that $P,N$ and $S$ are collinear.