Let $p$ be a prime number, and $X$ be the set of cubes modulo $p$, including $0$. Denote by $C_2(k)$ the number of ordered pairs $(x, y) \in X \times X$ such that $x + y \equiv k \pmod p$. Likewise, denote by $C_3(k)$ the number of ordered pairs $(x, y, z) \in X \times X \times X$ such that $x + y + z \equiv k \pmod p$. Prove that there are integers $a, b$ such that for all $k$ not in $X$, we have \[ C_3(k) = a\cdot C_2(k) + b. \] Proposed by Murilo Corato, Brazil.