USAJMO 2012 Problem 1

Given a triangle $ABC$ , let $P$ and $Q$ be points on segments $\overline{AB}$ and $\overline{AC}$ , respectively, such that $AP=AQ$ . Let $S$ and $R$ be distinct points on segment $\overline{BC}$ such that $S$ lies between $B$ and $R$ , $\angle BPS=\angle PRS$ , and $\angle CQR=\angle QSR$ . Prove that $P,Q,R,S$ are concyclic (in other words, these four points lie on a circle).