2009 USAMO 第 6 题

Let $s_1, s_2, s_3, \ldots$ be an infinite, nonconstant sequence of rational numbers, meaning it is not the case that $s_1 = s_2 = s_3 = \cdots.$ Suppose that $t_1, t_2, t_3, \ldots$ is also an infinite, nonconstant sequence of rational numbers with the property that $(s_i - s_j)(t_i - t_j)$ is an integer for all $i$ and $j$. Prove that there exists a rational number $r$ such that $(s_i - s_j)r$ and $(t_i - t_j)/r$ are integers for all $i$ and $j$.

设 $s_1, s_2, s_3, \ldots$ 是一个无限的、非常量的有理数序列，这意味着 $s_1 = s_2 = s_3 = \cdots$ 的情况并非如此。$假设$t_1, t_2, t_3, \ldots$也是一个无限的、非常量的有理数序列，其属性为$(s_i - s_j)(t_i - t_j)$是所有$i$和$j$的整数。证明存在一个有理数$r$，使得$(s_i - s_j)r$和$(t_i - t_j)/r$对于所有$i$和$j$ 都是整数。