SHARYGIN 2023 Problem 12

Let $ABC$ be a triangle with obtuse angle $B$ , and $P, Q$ lie on $AC$ in such a way that $AP = PB, BQ = QC$ . The circle $BPQ$ meets the sides $AB$ and $BC$ at points $N$ and $M$ respectively. $\textbf{(a)}$ (grades 8-9) Prove that the distances from the common point $R$ of $PM$ and $NQ$ to $A$ and $C$ are equal. $\textbf{(b)}$ (grades 10-11) Let $BR$ meet $AC$ at point $S$ . Prove that $MN \perp OS$ , where $O$ is the circumcenter of $ABC$ .