JBMO 2026 Problem 4

Let $ABC$ be a triangle with $AB \neq AC$ and let $I$ be its incenter. Let $P$ and $Q$ be points inside triangle $ABC$ such that $PB = PC > QC = QB$. Lines $BP$ and $CQ$ meet at $X$. Suppose that line $AI$ is tangent to the circumcircle of triangle $IPQ$. Prove that the circumcircles of triangles $ABQ, ACP$ and $PQX$ have a common point.