2012 IMO Shortlist A4

Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\operatorname{deg} f>\operatorname{deg} g$. Suppose that for infinitely many primes $p$ the polynomial $p f+g$ has a rational root. Prove that $f$ has a rational root.

令 $f$ 和 $g$ 为两个具有整数系数的非零多项式，且 $\operatorname{deg} f>\operatorname{deg} g$。假设对于无穷多个素数 $p$，多项式 $p f+g$ 具有有理根。证明 $f$ 有有理根。