KZ-MO 2017 Problem 1

The non-isosceles triangle $ABC$ is inscribed in the circle $\omega$ . The tangent line to this circle at the point $C$ intersects the line $AB$ at the point $D$ . Let the bisector of the angle $CDB$ intersect the segments $AC$ and $BC$ at the points $K$ and $L$ , respectively. The point $M$ is on the side $AB$ such that $\frac{AK}{BL} = \frac{AM}{BM}$ . Let the perpendiculars from the point $M$ to the straight lines $KL$ and $DC$ intersect the lines $AC$ and $DC$ at the points $P$ and $Q$ respectively. Prove that $2\angle CQP=\angle ACB$