International Mathematical Olympiad 2016 Problem 3

Let $P = A_1, A_2 \dots A_k$ be a convex polygon on the plane. The vertices $P = A_1, A_2 \dots A_k$ 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 $n$.

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