MathNet 2020 — Math Olympiad Vault — Part 03 Question 121

Let $\triangle ABC$ be an acute triangle with incentre $I$ and orthocentre $H$. $AI$ meets the circumcircle of $\triangle ABC$ again at $M$. Suppose the length $IM$ is exactly the circumradius of $\triangle ABC$. Show that $AH \ge AI$.