2012 USAJMO Problems/Problem 1

Problem

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).

2012 USAJMO (Problems • Resources)
First Problem	Followed by Problem 2
1 • 2 • 3 • 4 • 5 • 6
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2012 USAJMO Problems/Problem 1

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