IMO 2009 Shortlist 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 necessari...
Category: Number Theory
Problem
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(x) = (x + a)(x + b). (a) Prove that there exist distinct positive integers a and b such that all the numbers P(1),P(2), ...,P(50) are balanced. (b) Prove that if P(n) is balanced for all positive integers n, then a = b.