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## Sufficient condition for a sequence to be an enumeration of integers

Let $a_1,a_2,...$ be a sequence if integers with infinitely many positive and negative terms. Suppose that for any $n$ the numbers $a_1,a_2,...,a_n$ have $n$ different remainders modulo $n$. Prove that each integer occurs exactly once in the given sequence.
IMO 2005 Problem 2

Consider $S_n=\{a_1,..,a_n\}$. Suppose that $|a_i-a_j|=d \geq n$ for some $i,j$. Then $a_i \equiv a_j \mod d$ in $S_d$; Contradiction. Therefore $|a_i-a_j| for every $i,j \leq n$.

Consider the greatest element of $S_n$. Then, since $a_i$ are distinct, it follows that $S_n$ must be a list of consecutive integers.

Since the initial sequence contains infinitely many positive and negative integers it follows that $(a_n)$ is an enumeration of the integers.