CN-TST 2019 Problem 19

Cyclic quadrilateral $ABCD$ has circumcircle $(O)$ . Points $M$ and $N$ are the midpoints of $BC$ and $CD$ , and $E$ and $F$ lie on $AB$ and $AD$ respectively such that $EF$ passes through $O$ and $EO=OF$ . Let $EN$ meet $FM$ at $P$ . Denote $S$ as the circumcenter of $\triangle PEF$ . Line $PO$ intersects $AD$ and $BA$ at $Q$ and $R$ respectively. Suppose $OSPC$ is a parallelogram. Prove that $AQ=AR$ .