Baltic Way 2019

Since $AP$ is the symmedian of $AXY$ and $P$ lies on the perpendicular bisector of $XY$, then $PX$ and $PY$ are tangent to the circumcircle of triangle $AXY$. Therefore, we can easily find that $\angle YAX=60^{\circ }$ or $\angle YAX=120^{\circ }$.

Since $B$, $C$, $X$, $Y$ lie on a circle, $XY$ is antiparallel to $BC$ in the angle $BAC$. So, the $A$-median of $ABC$ is the $A$-symmedian of $AXY$. Hence $PX$, $PY$ are tangent to the circumcircle of $AXY$. Let $O$ be the circumcenter of $AXY$.

If $A$ and $P$ lie on different sides of the line $XY$ then we obtain $$\angle XAY=\frac{1}{2}\angle XOY=\frac{1}{2}(180^{\circ }-\angle YPX)=\frac{1}{2}(180^{\circ }-60^{\circ })=60^{\circ }.$$ If $A$ and $P$ lie on the same side of the line $XY$ then $\angle XAY=180^{\circ }-\frac{1}{2}\angle XOY=180^{\circ }-\frac{1}{2}\cdot 120^{\circ }=120^{\circ }$.

Hence, there are only two possible values of angle $XAY$: $60^{\circ }$ and $120^{\circ }$.