APMO 2013 Problem 3

For $2 k$ real numbers $a_{1}, a_{2}, \ldots, a_{k}, b_{1}, b_{2}, \ldots, b_{k}$ define the sequence of numbers $X_{n}$ by

$$X_{n}=\sum_{i=1}^{k}\left[a_{i} n+b_{i}\right] (n=1,2, \ldots)$$

If the sequence $X_{n}$ forms an arithmetic progression, show that $\sum_{i=1}^{k} a_{i}$ must be an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.