63rd International Mathematical Olympiad - IMO 2022

Problem 4.

Let $ABCDE$ be a convex pentagon such that $BC=DE$ . Assume that there is a point $T$ inside $ABCDE$ with $TB=TD,TC=TE$ and $\angle ABT = \angle TEA$ . Let line $AB$ intersect lines $CD$ and $CT$ at points $P$ and $Q$ , respectively. Assume that the points $P,B,A,Q$ occur on their line in that order. Let line $AE$ intersect $CD$ and $DT$ at points $R$ and $S$ , respectively. Assume that the points $R,E,A,S$ occur on their line in that order. Prove that the points $P,S,Q,R$ lie on a circle.