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IMO Shortlist 2009 problem N1

Let $n$ be a positive integer and let $a_1$ , $a_2$ , $a_3$ , ..., $a_k$ ( $k \geqslant 2$ ) be distinct integers in the set $\left\{1,\,2,\,\ldots,\,n\right\}$ such that $n$ divides $a_i \left(a_{i + 1} - 1\right)$ for $i = 1,\,2,\,\ldots,\,k - 1$ . Prove that $n$ does not divide $a_k \left(a_1 - 1\right)$ .

Proposed by Ross Atkins, Australia

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Zadaci

IMO Shortlist 1998 problem N1

Determine all pairs $(x,y)$ of positive integers such that $x^{2}y+x+y$ is divisible by $xy^{2}+y+7$ .

IMO Shortlist 1999 problem N1

Find all the pairs of positive integers $(x,p)$ such that p is a prime, $x \leq 2p$ and $x^{p-1}$ is a divisor of $(p-1)^{x}+1$ .

IMO Shortlist 2002 problem N2

Let $n\geq2$ be a positive integer, with divisors $1=d_1<d_2<\,\ldots<d_k=n$ . Prove that $d_1d_2+d_2d_3+\,\ldots\,+d_{k-1}d_k$ is always less than $n^2$ , and determine when it is a divisor of $n^2$ .

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 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 $a_1,a_2,\ldots$ .

IMO Shortlist 2006 problem N1

Determine all pairs $(x, y)$ of integers such that $1+2^{x}+2^{2x+1}= y^{2}.$