IMO Shortlist 1970 problem 1


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2. travnja 2012.
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Consider a regular 2n-gon and the n diagonals of it that pass through its center. Let P be a point of the inscribed circle and let a_1, a_2, \ldots , a_n be the angles in which the diagonals mentioned are visible from the point P. Prove that
\sum_{i=1}^n \tan^2 a_i = 2n \frac{\cos^2 \frac{\pi}{2n}}{\sin^4 \frac{\pi}{2n}}.
Izvor: Međunarodna matematička olimpijada, shortlist 1970