Question 110
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.
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.