USAJMO 2014 Question 2

Let $ABC$ be an acute nonequilateral triangle with $\angle A=60^\circ$, circumcenter $O$ and orthocenter $H$. (a) Prove that line $OH$ meets segments $AB,AC$ at $P,Q$, respectively. (b) If $s=[APQ]$ and $t=[BPQC]$, determine the range of $s/t$.

Proposed by Zuming Feng (credited in the supplied notes).