IMO 1982 LL BUL13

Origin: BUL

Problem

A regular n-gonal truncated pyramid is circumscribed around a sphere. Denote the areas of the base and the lateral surfaces of the pyramid by S1, S2, and S, respectively. Let \sigma be the area of the polygon whose vertices are the tangential points of the sphere and the lateral faces of the pyramid. Prove that \sigmaS = 4S1S2 cos2 \pi n.