2001 APMO 第 1 题

Two equal-sized regular $n$ -gons intersect to form a $2n$ -gon $C$ . Prove that the sum of the sides of $C$ which form part of one $n$ -gon equals half the perimeter of $C$ .

*Alternative formulation:*

Let two equal regular $n$ -gons $S$ and $T$ be located in the plane such that their intersection $S\cap T$ is a $2n$ -gon (with $n\ge 3$ ). The sides of the polygon $S$ are coloured in red and the sides of $T$ in blue.

Prove that the sum of the lengths of the blue sides of the polygon $S\cap T$ is equal to the sum of the lengths of its red sides.

两个大小相等的常规 $n$ 边形相交形成 $2n$ 边形 $C$ 。证明构成 $n$ 边形一部分的 $C$ 的边的总和等于 $C$ 周长的一半。

*替代配方：*

让两个相等的规则$n$边形$S$和$T$位于平面中，使得它们的交集$S\cap T$是$2n$边形(与$n\ge 3$)。多边形 $S$ 的边为红色，$T$ 的边为蓝色。

证明多边形 $S\cap T$ 的蓝色边的长度之和等于其红色边的长度之和。