2024 IMO Problems/Problem 4

Let $ABC$ be a triangle with $AB < AC < BC$ . Let the incentre and incircle of triangle $ABC$ be $I$ and $\omega$ , respectively. Let $X$ be the point on line $BC$ different from $C$ such that the line through $X$ parallel to $AC$ is tangent to $\omega$ . Similarly, let $Y$ be the point on line $BC$ different from $B$ such that the line through $Y$ parallel to $AB$ is tangent to $\omega$ . Let $AI$ intersect the circumcircle of triangle $ABC$ again at $P \neq A$ . Let $K$ and $L$ be the midpoints of $AC$ and $AB$ , respectively. Prove that $\angle KIL + \angle YPX = 180^{\circ}$ .