2009 APMO 第 4 题

Prove that for any positive integer $k$, there exists an arithmetic sequence

$$\frac{a_{1}}{b_{1}}, \quad \frac{a_{2}}{b_{2}}, \ldots, \quad \frac{a_{k}}{b_{k}}$$

of rational numbers, where $a_{i}, b_{i}$ are relatively prime positive integers for each $i=1,2, \ldots, k$, such that the positive integers $a_{1}, b_{1}, a_{2}, b_{2}, \ldots, a_{k}, b_{k}$ are all distinct.

证明对于任意正整数$k$，存在等差数列

$$\frac{a_{1}}{b_{1}}、\quad \frac{a_{2}}{b_{2}}、\ldots、\quad \frac{a_{k}}{b_{k}}$$

有理数，其中 $a_{i}, b_{i}$ 是每个 $i=1,2, \ldots, k$ 的相对质数正整数，使得正整数 $a_{1}, b_{1}, a_{2}, b_{2}, \ldots, a_{k}, b_{k}$ 都是不同的。