2011 IMO Shortlist N2

Consider a polynomial $P(x)=\left(x+d_{1}\right)\left(x+d_{2}\right) \cdot \ldots \cdot\left(x+d_{9}\right)$, where $d_{1}, d_{2}, \ldots, d_{9}$ are nine distinct integers. Prove that there exists an integer $N$ such that for all integers $x \geq N$ the number $P(x)$ is divisible by a prime number greater than 20 .

考虑一个多项式 $P(x)=\left(x+d_{1}\right)\left(x+d_{2}\right) \cdot \ldots \cdot\left(x+d_{9}\right)$，其中 $d_{1}、d_{2}、\ldots、d_{9}$ 是九个不同的整数。证明存在一个整数 $N$ ，使得对于所有整数 $x \geq N$ ，数字 $P(x)$ 可以被大于 20 的素数整除。