CN-TST 2021 Problem 4

Let $f(x),g(x)$ be two polynomials with integer coefficients. It is known that for infinitely many prime $p$ , there exist integer $m_p$ such that $$f(a) \equiv g(a+m_p) \pmod p$$ holds for all $a \in \mathbb{Z}.$ Prove that there exists a rational number $r$ such that $$f(x)=g(x+r).$$