IR-TST 2012 Problem 11

Points $A$ and $B$ are on a circle $\omega$ with center $O$ such that $\tfrac{\pi}{3}< \angle AOB <\tfrac{2\pi}{3}$ . Let $C$ be the circumcenter of the triangle $AOB$ . Let $l$ be a line passing through $C$ such that the angle between $l$ and the segment $OC$ is $\tfrac{\pi}{3}$ . $l$ cuts tangents in $A$ and $B$ to $\omega$ in $M$ and $N$ respectively. Suppose circumcircles of triangles $CAM$ and $CBN$ , cut $\omega$ again in $Q$ and $R$ respectively and theirselves in $P$ (other than $C$ ). Prove that $OP\perp QR$ .

Proposed by Mehdi E'tesami Fard, Ali Khezeli