Olympiad problems

2014 ELMO Shortlist, 7

Let $ABC$ be a triangle inscribed in circle $\omega$ with center $O$, let $\omega_A$ be its $A$-mixtilinear incircle, $\omega_B$ be its $B$-mixtilinear incircle, $\omega_C$ be its $C$-mixtilinear incircle, and $X$ be the radical center of $\omega_A$, $\omega_B$, $\omega_C$. Let $A'$, $B'$, $C'$ be the points at which $\omega_A$, $\omega_B$, $\omega_C$ are tangent to $\omega$. Prove that $AA'$, $BB'$, $CC'$ and $OX$ are concurrent. [i]Proposed by Robin Park[/i]

2014 ELMO Shortlist, 7

Tags: incircle , geometry , concurrency , mixtilinear circle , mixtilinear incircle

Let $ABC$ be a triangle inscribed in circle $\omega$ with center $O$, let $\omega_A$ be its $A$-mixtilinear incircle, $\omega_B$ be its $B$-mixtilinear incircle, $\omega_C$ be its $C$-mixtilinear incircle, and $X$ be the radical center of $\omega_A$, $\omega_B$, $\omega_C$. Let $A'$, $B'$, $C'$ be the points at which $\omega_A$, $\omega_B$, $\omega_C$ are tangent to $\omega$. Prove that $AA'$, $BB'$, $CC'$ and $OX$ are concurrent. [i]Proposed by Robin Park[/i]

2016 IMO Shortlist, G2

Tags: projective geometry , mixtilinear incircle , median , geometry

Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$. The points $D$, $E$, $F$ are selected on sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ such that $\overline{ID} \perp \overline{BC}$, $\overline{IE}\perp \overline{AI}$, and $\overline{IF}\perp \overline{AI}$. Suppose that the circumcircle of $\triangle AEF$ intersects $\Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $\Gamma$. [i]Proposed by Evan Chen, Taiwan[/i]