Let $H$ be the orthocenter and $O$ the circumcenter of the acute triangle $\vartriangle ABC$. The circle with center $H$ and radius $HA$ intersects the lines $AC$ and $AB$ at points $P$ and $Q$, respectively. Let point $O$ be the orthocenter of triangle $\vartriangle APQ$, determine the measure of $\angle BAC$.
Problem
Source: OLCOMA Costa Rica National Olympiad, Final Round, 2018 Shortlist G1 day1
Tags: geometry, Circumcenter, orthocenter, angles