2022 IMO Shortlist N7

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 (modulo rotation and reflection) to place the elements of $S$ around a circle such that the product of any two neighbors is of the form $x^{2}+x+k$ for some positive integer $x$. (U.S.A.)

设 $k$ 为正整数，$S$ 为奇质数的有限集。证明最多有一种方法(模旋转和反射)将 $S$ 的元素放置在一个圆周围，使得对于某个正整数 $x$，任何两个邻居的乘积的形式为 $x^{2}+x+k$。 (美国。)