US-TST 2005 Problem 6

Let $ABC$ be an acute scalene triangle with $O$ as its circumcenter. Point $P$ lies inside triangle $ABC$ with $\angle PAB = \angle PBC$ and $\angle PAC = \angle PCB$ . Point $Q$ lies on line $BC$ with $QA = QP$ . Prove that $\angle AQP = 2\angle OQB$ .