1970 IMO 第 1 题

(BEL 3) Consider a regular $2 n$-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 $$\sum_{i=1}^{n} \tan ^{2} a_{i}=2 n \frac{\cos ^{2} \frac{\pi}{2 n}}{\sin ^{4} \frac{\pi}{2 n}}$$

(BEL 3) 考虑一个规则的 $2 n$ 边形及其穿过其中心的 $n$ 对角线。令$P$为内切圆的一点，并令$a_{1}、a_{2}、\ldots、a_{n}$为从点$P$可见的对角线的角度。证明 $$\sum_{i=1}^{n} \tan ^{2} a_{i}=2 n \frac{\cos ^{2} \frac{\pi}{2 n}}{\sin ^{4} \frac{\pi}{2 n}}$$