HMIC 2022 Problem 5

Let $p$ be a prime and let $\mathbb{F}_{p}$ be the set of integers modulo $p$. Call a function $f: \mathbb{F}_{p}^{2} \rightarrow \mathbb{F}_{p}$ quasiperiodic if there exist $a, b \in \mathbb{F}_{p}$, not both zero, so that $f(x+a, y+b)=f(x, y)$ for all $x, y \in \mathbb{F}_{p}$. Find, with proof, the number of functions $\mathbb{F}_{p}^{2} \rightarrow \mathbb{F}_{p}$ that can be written as the sum of some number of quasiperiodic functions. Note: The original version of this problem neglected to mention that p was a prime. We regret the error.