USAJMO 2014 Problem 2

Let $\triangle{ABC}$ be a non-equilateral, acute triangle with $\angle A=60^\circ$ , and let $O$ and $H$ denote the circumcenter and orthocenter of $\triangle{ABC}$ , respectively.

(a) Prove that line $OH$ intersects both segments $AB$ and $AC$ .

(b) Line $OH$ intersects segments $AB$ and $AC$ at $P$ and $Q$ , respectively. Denote by $s$ and $t$ the respective areas of triangle $APQ$ and quadrilateral $BPQC$ . Determine the range of possible values for $s/t$ .