DANUBE 2015 Problem 4

Let $ABCD$ be a rectangle with $AB\ge BC$ Point $M$ is located on the side $(AD)$ , and the perpendicular bisector of $[MC]$ intersects the line $BC$ at the point $N$ . Let ${Q} =MN\cup AB$ . Knowing that $\angle MQA= 2\cdot \angle BCQ$ , show that the quadrilateral $ABCD$ is a square.