APMO 2009 Problem 4

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

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

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.