RMO 2025 Problem 3

Let $\Omega$ and $\Gamma$ be circles centred at $O_1$ , $O_2$ respectively. Suppose that they intersect in distinct points $A$ , $B$ . Suppose $O_1$ is outside $\Gamma$ and $O_2$ is outside $\Omega$ . Let $l$ be not passing through $A$ and $B$ that interests $\Omega$ at $P$ and $R$ and $\Gamma$ at $Q$ , $S$ so that $P$ , $Q$ , $R$ , $S$ lie on the line in this order. Furthermore, points $O_1$ , $B$ lie on one side of $l$ and the points $O_2$ , $A$ lie on the other side of $l$ . Given that the points $A$ , $P$ , $Q$ , $O_1$ are concyclic and $B$ , $R$ , $S$ , $O_2$ are concyclic as well, prove that $AQ=BR$ Proposed by Archit Manas and Aadish Jain