USAMO 2016 — Days 1 and 2 Question 5

An equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M \in \overline{AB}$, $Q \in \overline{AC}$, and $N,P \in \overline{BC}$. Let $S$ be the intersection of $\overline{MN}$ and $\overline{PQ}$. Denote by $\ell$ the angle bisector of $\angle MSQ$. Prove that $\overline{OI}$ is parallel to $\ell$, where $O$ is the circumcenter of triangle $ABC$, and $I$ is the incenter of triangle $ABC$.

Proposed by Ivan Borsenco (credited by Evan Chen).