ELMO SL 2011 G4

Prove that for any convex pentagon $A_1A_2A_3A_4A_5$ , there exists a unique pair of points $\{P,Q\}$ (possibly with $P=Q$ ) such that $\measuredangle{PA_i A_{i-1}} = \measuredangle{A_{i+1}A_iQ}$ for $1\le i\le 5$ , where indices are taken $\pmod5$ and angles are directed $\pmod\pi$ .

Calvin Deng.