IGO 2020 I1

A trapezoid $ABCD$ is given where $AB$ and $CD$ are parallel. Let $M$ be the midpoint of the segment $AB$ . Point $N$ is located on the segment $CD$ such that $\angle ADN = \frac{1}{2} \angle MNC$ and $\angle BCN = \frac{1}{2} \angle MND$ . Prove that $N$ is the midpoint of the segment $CD$ .

Proposed by Alireza Dadgarnia