2009 IMO Shortlist N2

PER 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 $P(1), P(2)$, $\ldots, P(50)$ are balanced. (b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$.

PER 如果 $N=1$ 或 $N$ 可以写为偶数个不一定不同素数的乘积，则正整数 $N$ 称为平衡的。给定正整数 $a$ 和 $b$，考虑由 $P(x)=(x+a)(x+b)$ 定义的多项式 $P$。 (a) 证明存在不同的正整数$a$和$b$，使得所有数字$P(1), P(2)$, $\ldots, P(50)$是平衡的。 (b) 证明如果$P(n)$ 对于所有正整数$n$ 是平衡的，则$a=b$。