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IMO Shortlist 2005 problem N2

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 $a_1,a_2,\ldots$ .

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IMO Shortlist 2005 problem N1

Determine all positive integers relatively prime to all the terms of the infinite sequence $a_n=2^n+3^n+6^n -1,\ n\geq 1.$

IMO Shortlist 2003 problem N3

Determine all pairs of positive integers $(a,b)$ such that $\dfrac{a^2}{2ab^2-b^3+1}$ is a positive integer.

IMO Shortlist 2003 problem N1

Let $m$ be a fixed integer greater than $1$ . The sequence $x_0$ , $x_1$ , $x_2$ , $\ldots$ is defined as follows:

$x_i= 2^i$ if $0 \leq i\leq m-1$ and $x_i = \sum_{j=1}^{m}x_{i-j},$ if $i\geq m$ .

Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by $m$ .