IMO Shortlist 2009 problem N2

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Dodao/la: arhiva
April 2, 2012
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
Source: Međunarodna matematička olimpijada, shortlist 2009