KöMaL 2011 A525

Let $f(x)=x^n+a_{n-1}x^{n-1} +\ldots+a_1x+a_0$ be a polynomial with integer coefficients, and let $d_1,\ldots,d_n$ be pairwise distinct integers. Suppose that for infinitely many prime numbers $p$ there exists an integer $k_p$ for which $f(k_p+d_1)\equiv f(k_p+d_2)\equiv\dots\equiv f(k_p+d_n)\equiv0\pmod{p}.$ Prove that there exists an integer $k_0$ such that $f(k_0+d_1)=f(k_0+d_2)=\ldots=f(k_0+d_n)=0.$