International Mathematical Olympiad 2001 Problem 5

In a triangle $ABC$, let $AP$ bisect $\angle BAC$, with $P$ on $BC$, and let $BQ$ bisect $\angle ABC$, with $Q$ on $CA$.

It is known that $\angle BAC = 60^{\circ}$ and that $AB + BP = AQ + QB$.

What are the possible angles of triangle $ABC$?