Let
be an odd prime and
a positive integer. In the coordinate plane, eight distinct points with integer coordinates lie on a circle with diameter of length
. Prove that there exists a triangle with vertices at three of the given points such that the squares of its side lengths are integers divisible by
.
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Let $p$ be an odd prime and $n$ a positive integer. In the coordinate plane, eight distinct points with integer coordinates lie on a circle with diameter of length $p^{n}$. Prove that there exists a triangle with vertices at three of the given points such that the squares of its side lengths are integers divisible by $p^{n+1}$.