IN-TST 2010 Problem 8

Call a positive integer good if either $N=1$ or $N$ can be written as product of even number of prime numbers, not necessarily distinct. Let $P(x)=(x-a)(x-b),$ where $a,b$ are positive integers.

(a) Show that there exist distinct positive integers $a,b$ such that $P(1),P(2),\cdots ,P(2010)$ are all good numbers. (b) Suppose $a,b$ are such that $P(n)$ is a good number for all positive integers $n$ . Prove that $a=b$ .