2004 IMO Shortlist S19

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 $p^{n+1}$ .

*Proposed by Alexander Ivanov, Bulgaria*

设$p$ 为奇素数，$n$ 为正整数。在坐标平面中，具有整数坐标的八个不同点位于直径为 $p^{n}$ 的圆上。证明存在一个三角形，其顶点位于给定的三个点处，使得其边长的平方是可被 $p^{n+1}$ 整除的整数。

*由保加利亚亚历山大·伊万诺夫提议*