Problem

Source: IX International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

Tags: number theory



The set of numbers $(p, a, b, c)$ of positive integers is called Sozopolian when: * p is an odd prime number * $a$, $b$ and $c$ are different and * $ab + 1$, $bc + 1$ and $ca + 1$ are a multiple of $p$. a) Prove that each Sozopolian set satisfies the inequality $p+2 \leq \frac{a+b+c}{3}$ b) Find all numbers $p$ for which there exist a Sozopolian set for which the equality of the upper inequation is met.