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Problem 4. Let $ABC$ be a triangle such that $AB \neq AC$ and let $I$ be the incenter of $\triangle ABC$. Let $P$ and $Q$ be points inside $\triangle ABC$ such that $PB = PC > QC = QB$. The lines $BP$ and $CQ$ intersects at point $X$. Assume that $AI$ is tangent to the circumcircle of $\triangle IPQ$. Prove that the circumcircles of the triangles $ABQ$, $ACP$, and $PQX$ have a point in common.
Solution 1Solution 2Solution 3Solution 4Solution 5
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