XK-TST 2022 Problem 3

Let $ABC$ be a triangle and $D$ point on side $BC$ such that $AD$ is angle bisector of angle $\angle BAC$ . Let $E$ be the intersection of the side $AB$ with circle $\omega_1$ which has diameter $CD$ and let $F$ be the intersection of the side $AC$ with circle $\omega_2$ which has diameter $BD$ . Suppose that there exist points $P\in\omega_1$ and $Q\in\omega_2$ such that $E, P, Q$ and $F$ are collinear and on this order. Prove that $AD, BQ$ and $CP$ are concurrent.

Proposed by Dorlir Ahmeti, Kosovo and Noah Walsh, U.S.A.