2005 IMO 第 2 题

Let $a_{1}$ , $a_{2}$ , ... be a sequence of integers with infinitely many positive and negative terms. Suppose that for every positive integer $n$ the numbers $a_{1}$ , $a_{2}$ , ..., $a_{n}$ leave $n$ different remainders upon division by $n$ . Prove that every integer occurs exactly once in the sequence.

令 $a_{1}$ 、 $a_{2}$ 、... 为具有无限多个正项和负项的整数序列。假设对于每个正整数 $n$，数字 $a_{1}$ 、 $a_{2}$ 、 ...、 $a_{n}$ 除以 $n$ 后留下 $n$ 不同的余数。证明每个整数在序列中恰好出现一次。