USAJMO 2024 Problem 1

Let $ABCD$ be a cyclic quadrilateral with $AB = 7$ and $CD = 8$ . Point $P$ and $Q$ are selected on segment $AB$ such that $AP = BQ = 3$ . Points $R$ and $S$ are selected on segment $CD$ such that $CR = DS = 2$ . Prove that $PQRS$ is a cyclic quadrilateral.

Proposed by Evan O'Dorney