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United States competition problems

Geometry Difficulty 5.5 AIME, harder Prove it United States

Problem:

Let $ABC$ be a triangle with incenter $I$ and circumcenter $O$ . Let the circumradius be $R$ . What is the least upper bound of all possible values of $IO$ ?

Solution

Solution:

$I$ always lies inside the convex hull of $ABC$ , which in turn always lies in the circumcircle of $ABC$ , so $IO < R$ . On the other hand, if we first draw the circle $\Omega$ of radius $R$ about $O$ and then pick $A$ , $B$ , and $C$ very close together on it, we can force the convex hull of $ABC$ to lie outside the circle of radius $R - \epsilon$ about $O$ for any $\epsilon$ . Thus the answer is $R$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.