OIM 2020 Problem 1

Let $ABC$ be an acute scalene triangle such that $AB <AC$ . The midpoints of sides $AB$ and $AC$ are $M$ and $N$ , respectively. Let $P$ and $Q$ be points on the line $MN$ such that $\angle CBP = \angle ACB$ and $\angle QCB = \angle CBA$ . The circumscribed circle of triangle $ABP$ intersects line $AC$ at $D$ ( $D\ne A$ ) and the circumscribed circle of triangle $AQC$ intersects line $AB$ at $E$ ( $E \ne A$ ). Show that lines $BC, DP,$ and $EQ$ are concurrent.

Nicolás De la Hoz, Colombia