2016 IMO 第 3 题

Let $P = A_{1}A_{2}\ldots A_{k}$ be a convex polygon in the plane. The vertices $A_{1}$ $A_{2}$ ,..., $A_{k}$ have integral coordinates and lie on a circle. Let $S$ be the area of $P$ . An odd positive integer $n$ is given such that the squares of the side lengths of $P$ are integers divisible by $n$ . Prove that $2S$ is an integer divisible by $n$ .

令 $P = A_{1}A_{2}\ldots A_{k}$ 为平面中的凸多边形。顶点 $A_{1}$ $A_{2}$ ,..., $A_{k}$ 具有积分坐标并且位于圆上。令 $S$ 为 $P$ 的面积。给出一个奇数正整数 $n$ ，使得 $P$ 的边长的平方是可被 $n$ 整除的整数。证明 $2S$ 是可被 $n$ 整除的整数。