IMO Shortlist 2005 problem N2
Let
be a sequence of integers with infinitely many positive and negative terms. Suppose that for every positive integer
the numbers
leave
different remainders upon division by
.
Prove that every integer occurs exactly once in the sequence
.
%V0
Let $a_1,a_2,\ldots$ be a sequence of integers with infinitely many positive and negative terms. Suppose that for every positive integer $n$ the numbers $a_1,a_2,\ldots,a_n$ leave $n$ different remainders upon division by $n$.
Prove that every integer occurs exactly once in the sequence $a_1,a_2,\ldots$.
Source: Međunarodna matematička olimpijada, shortlist 2005
Školjka 
