2013 APMO 第 3 题

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

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

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 $r$.

对于 $2 k$ 实数 $a_{1}, a_{2}, \ldots, a_{k}, b_{1}, b_{2}, \ldots, b_{k}$ 定义数字序列 $X_{n}$

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

如果序列 $X_{n}$ 形成算术级数，则证明 $\sum_{i=1}^{k} a_{i}$ 必须是整数。这里$[r]$表示小于或等于$r$的最大整数。