MathNet 2024 — batch of 27 questions Question 17

Suppose $p$ is a prime number and $x, y, z$ are integers satisfying $0 \lt x \lt y \lt z \lt p$. If $x^3, y^3, z^3$ have equal remainders when divided by $p$, prove that $x^2 + y^2 + z^2$ is divisible by $x + y + z$.