HMMT February

Geometry Difficulty 5.0 AIME Find the answer

Let $A B C$ 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 $I O$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

$I$ always lies inside the convex hull of $A B C$ , which in turn always lies in the circumcircle of $A B C$ , so $I O<R$ . 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 $A B C$ to lie outside the circle of radius $R-\epsilon$ about $O$ for any $\epsilon$ . Thus the answer is $R$ .

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