2016 IMO Shortlist N7

Let $n$ be an odd positive integer. In the Cartesian plane, a cyclic polygon $P$ with area $S$ is chosen. All its vertices have integral coordinates, and all squares of its side lengths are divisible by $n$. Prove that $2 S$ is an integer divisible by $n$.

令 $n$ 为奇数正整数。在笛卡尔平面中，选择面积为 $S$ 的循环多边形 $P$。它的所有顶点都有整数坐标，并且其边长的所有正方形都可以被 $n$ 整除。证明 $2 S$ 是可被 $n$ 整除的整数。