Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.8 AIME Prove it United States

Problem:

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 $C$ .

Solution

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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