Problem

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2016 Shortlist N2 day2

Tags: divides, divisible, number theory



Let $x, y, z$ be positive integers and $p$ a prime such that $x <y <z <p$. Also $x^3, y^3, z^3$ leave the same remainder when divided by $p$. Prove that $x + y + z$ divides $x^2 + y^2 + z^2$.