2022 IMO 第 3 题

Let $k$ be a positive integer and let $S$ be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of $S$ around the circle such that the product of any two neighbors is of the form $x^{2} + x + k$ for some positive integer $x$ .

设 $k$ 为正整数，$S$ 为奇质数的有限集。证明最多有一种方法(最多旋转和反射)将 $S$ 的元素放置在圆周围，使得对于某个正整数 $x$ ，任何两个邻居的乘积的形式为 $x^{2} + x + k$ 。