ELMO 2014 Problem 5

Let $ABC$ be a triangle with circumcenter $O$ and orthocenter $H$ . Let $\omega_1$ and $\omega_2$ denote the circumcircles of triangles $BOC$ and $BHC$ , respectively. Suppose the circle with diameter $\overline{AO}$ intersects $\omega_1$ again at $M$ , and line $AM$ intersects $\omega_1$ again at $X$ . Similarly, suppose the circle with diameter $\overline{AH}$ intersects $\omega_2$ again at $N$ , and line $AN$ intersects $\omega_2$ again at $Y$ . Prove that lines $MN$ and $XY$ are parallel.

Proposed by Sammy Luo