IMO 1985 LL TUR83

Let \Gammai, i = 0, 1, 2, . . ., be a circle of radius ri inscribed in an

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IMO 1985 LL TUR83

Origin: TUR

Problem

Let \Gammai, i = 0, 1, 2, . . ., be a circle of radius ri inscribed in an angle of measure 2\alpha such that each \Gammai is externally tangent to \Gammai+1 and ri+1 < ri. Show that the sum of the areas of the circles \Gammai is equal to the area of a circle of radius r = 1 2r0( \sqrt sin \alpha + \sqrtcsc \alpha).