2018 USAMO 第 6 题

Let $a_n$ be the number of permutations $(x_1, x_2, \dots, x_n)$ of the numbers $(1,2,\dots, n)$ such that the $n$ ratios $\frac{x_k}{k}$ for $1\le k\le n$ are all distinct. Prove that $a_n$ is odd for all $n\ge 1.$

令$a_n$为数字$(1,2,\dots, n)$的排列数$(x_1, x_2, \dots, x_n)$，使得$1\le k\le n$的$n$比率$\frac{x_k}{k}$都是不同的。证明 $a_n$ 对于所有 $n\ge 1.$ 都是奇数