IMO 1989 LL KOR64

Let a regular (2n + 1)-gon be inscribed in a circle of radius r.

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IMO 1989 LL KOR64

Origin: KOR

Problem

Let a regular (2n + 1)-gon be inscribed in a circle of radius r. We consider all the triangles whose vertices are from those of the regular (2n + 1)-gon. (a) How many triangles among them contain the center of the circle in their interior? (b) Find the sum of the areas of all those triangles that contain the center of the circle in their interior.