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IMO Shortlist 2009 problem N2

A positive integer $N$ is called balanced, if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$ , consider the polynomial $P$ defined by $P\!\left(x\right) = \left(x+a\right)\left(x+b\right)$ .
a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P\!\left(1\right)$ , $P\!\left(2\right)$ , ..., $P\!\left(50\right)$ are balanced.
b) Prove that if $P\!\left(n\right)$ is balanced for all positive integers $n$ , then $a=b$ .

Proposed by Jorge Tipe, Peru

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IMO Shortlist 1993 problem N1

A natural number $n$ is said to have the property $P,$ if, for all $a, n^2$ divides $a^n - 1$ whenever $n$ divides $a^n - 1.$

a.) Show that every prime number $n$ has property $P.$

b.) Show that there are infinitely many composite numbers $n$ that possess property $P.$

IMO Shortlist 1993 problem N2

Let $a,b,n$ be positive integers, $b > 1$ and $b^n-1|a.$ Show that the representation of the number $a$ in the base $b$ contains at least $n$ digits different from zero.

IMO Shortlist 2008 problem N1

Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$ , $b$ , $c$ are integers (not necessarily positive) satisfying the equations
$a^n + pb = b^n + pc = c^n + pa$
then $a = b = c$ .

Proposed by Angelo Di Pasquale, Australia

IMO Shortlist 2008 problem N2

Let $a_1$ , $a_2$ , $\ldots$ , $a_n$ be distinct positive integers, $n\ge 3$ . Prove that there exist distinct indices $i$ and $j$ such that $a_i + a_j$ does not divide any of the numbers $3a_1$ , $3a_2$ , $\ldots$ , $3a_n$ .

Proposed by Mohsen Jamaali, Iran

IMO Shortlist 2009 problem N4

Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$ , $a_2$ , ..., $a_n$ satisfying
$a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1$ for every $k$ with $2 \leqslant k \leqslant n-1$ .

Proposed by North Korea

MEMO 2009 ekipno problem 8

Find all non-negative integer solutions of the equation $2^x+2009=3^y5^z.$