US-TST 2011 Problem 7

Let $ABC$ be an acute scalene triangle inscribed in circle $\Omega$ . Circle $\omega$ , centered at $O$ , passes through $B$ and $C$ and intersects sides $AB$ and $AC$ at $E$ and $D$ , respectively. Point $P$ lies on major arc $BAC$ of $\Omega$ . Prove that lines $BD, CE, OP$ are concurrent if and only if triangles $PBD$ and $PCE$ have the same incenter.