ELMO 2024 Problem 6

For a prime $p$ , let $\mathbb{F}_p$ denote the integers modulo $p$ , and let $\mathbb{F}_p[x]$ be the set of polynomials with coefficients in $\mathbb{F}_p$ . Find all $p$ for which there exists a quartic polynomial $P(x) \in \mathbb{F}_p[x]$ such that for all integers $k$ , there exists some integer $\ell$ such that $P(\ell) \equiv k \pmod p$ . (Note that there are $p^4(p-1)$ quartic polynomials in $\mathbb{F}_p[x]$ in total.)

Aprameya Tripathy