Question 34
Let $P = A_1A_2 \ldots A_k$ be a convex polygon in the plane. The vertices $A_1, A_2, \ldots, A_k$ have integer coordinates and lie on a circle. Let $S$ be the area of $P$. Let $n$ be an odd positive integer such that the squares of the side lengths of $P$ are all integers divisible by $n$. Prove that $2S$ is an integer divisible by $n$.