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For a prime number $p$ and a polynomial $f$ with integer coefficients, define $\text{Im}(p,f)$ be the set of integers $a\in \{0,1,\dots ,p-1\}$ such that there exists an integer $x$, for which $f(x)-a$ is divisible by $p$.

Prove that there exist nonconstant polynomials $f$ and $g$ such that, for infinitely many primes, the intersection of $\text{Im}(p,f)$ and $\text{Im}(p,g)$ is empty.