TSTST 2011 Problem 4

Acute triangle $A B C$ is inscribed in circle $\omega$. Let $H$ and $O$ denote its orthocenter and circumcenter, respectively. Let $M$ and $N$ be the midpoints of sides $A B$ and $A C$, respectively. Rays $M H$ and $N H$ meet $\omega$ at $P$ and $Q$, respectively. Lines $M N$ and $P Q$ meet at $R$. Prove that $\overline{O A} \perp \overline{R A}$.