USAJMO 2017 Question 5

Let $O,H$ be the circumcenter and orthocenter of acute triangle $ABC$. Points $M,D$ lie on $BC$ with $BM=CM$ and $\angle BAD=\angle CAD$. Ray $MO$ meets the circumcircle of $BHC$ at $N$. Prove that $\angle ADO=\angle HAN$.

Proposed by Ivan Borsenco.