CMO 2025 Problem 4

Let $ABC$ be a triangle with circumcircle $\Gamma$ and $AB \neq AC$ . Let $D$ and $E$ lie on the arc $BC$ of $\Gamma$ not containing $A$ such that $\angle BAE = \angle DAC$ . Let the incenters of $BAE$ and $CAD$ be $X$ and $Y$ respectively, and let the external tangents of the incircles of $BAE$ and $CAD$ intersect at $Z$ . Prove that $Z$ lies on the common chord of $\Gamma$ and the circumcircle of $AXY$ .