IMO Shortlist 2005 problem N2
Dodao/la:
arhiva2. travnja 2012. Let
![a_1,a_2,\ldots](/media/m/d/b/c/dbce63436fd54e80a8e6c1712c9f50ca.png)
be a sequence of integers with infinitely many positive and negative terms. Suppose that for every positive integer
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
the numbers
![a_1,a_2,\ldots,a_n](/media/m/e/1/c/e1cadd08528b76b10be041b63c00aa8b.png)
leave
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
different remainders upon division by
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
.
Prove that every integer occurs exactly once in the sequence
![a_1,a_2,\ldots](/media/m/d/b/c/dbce63436fd54e80a8e6c1712c9f50ca.png)
.
%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 $a_1,a_2,\ldots$.
Izvor: Međunarodna matematička olimpijada, shortlist 2005