MathDB
IMO Shortlist 2009 - Problem N2

Source:

July 5, 2010
algebrapolynomialnumber theoryIMO Shortlist

Problem Statement

A positive integer NN is called balanced, if N=1N=1 or if NN can be written as a product of an even number of not necessarily distinct primes. Given positive integers aa and bb, consider the polynomial PP defined by P(x)=(x+a)(x+b)P(x)=(x+a)(x+b). (a) Prove that there exist distinct positive integers aa and bb such that all the number P(1)P(1), P(2)P(2),…\ldots, P(50)P(50) are balanced. (b) Prove that if P(n)P(n) is balanced for all positive integers nn, then a=ba=b.
Proposed by Jorge Tipe, Peru
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