US-TST 2006 Problem 6

Let $ABC$ be a triangle. Triangles $PAB$ and $QAC$ are constructed outside of triangle $ABC$ such that $AP = AB$ and $AQ = AC$ and $\angle{BAP}= \angle{CAQ}$ . Segments $BQ$ and $CP$ meet at $R$ . Let $O$ be the circumcenter of triangle $BCR$ . Prove that $AO \perp PQ.$