Question 1
Let $ABC$ be a triangle with $\angle A\lt 60^\circ$. Let $X,Y$ be points on $AB,AC$, respectively, such that $CA+AX=CB+BX$ and $BA+AY=BC+CY$. Let $P$ be the point in the plane such that $PX\perp AB$ and $PY\perp AC$. Prove that $\angle BPC\lt 120^\circ$.
