2008 Mock ARML 1 Problems/Problem 8
For positive real numbers ,
We consider a geometric interpretation, specifically with an equilateral triangle. Let the distances from the vertices to the incenter be , , and , and the tangents to the incircle be , , and . Then use Law of Cosines to express the sides in terms of , , and , and Pythagorean Theorem to express , , and in terms of , , , and the inradius . This yields the first three equations. The fourth is the result of the sine area formula for the three small triangles, and gives the area as . The desired expression is , which is also the area, so the answer is .
Since the equations are symmetric in , we may consider ; the system reduces and we find that the desired sum is .
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