IR-TST 2022 Problem 4

Cyclic quadrilateral $ABCD$ with circumcenter $O$ is given. Point $P$ is the intersection of diagonals $AC$ and $BD$ . Let $M$ and $N$ be the midpoint of the sides $AD$ and $BC$ , respectively. Suppose that $\omega_1$ , $\omega_2$ and $\omega_3$ be the circumcircle of triangles $ADP$ , $BCP$ and $OMN$ , respectively. The intersection point of $\omega_1$ and $\omega_3$ , which is not on the arc $APD$ of $\omega_1$ , is $E$ and the intersection point of $\omega_2$ and $\omega_3$ , which is not on the arc $BPC$ of $\omega_2$ , is $F$ . Prove that $OF=OE$ .

Proposed by Seyed Amirparsa Hosseini Nayeri