Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.9 AIME Find the answer

Let $C$ be a circle with two diameters intersecting at an angle of 30 degrees. A circle $S$ is tangent to both diameters and to $C$ , and has radius 1. Find the largest possible radius of $C$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

For $C$ to be as large as possible we want $S$ to be as small as possible. It is not hard to see that this happens in the situation shown below. Then the radius of $C$ is $1+\csc 15=\mathbf{1}+\sqrt{\mathbf{2}}+\sqrt{\mathbf{6}}$ . The computation of $\sin 15$ can be done via the half angle formula.

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