Problem

Source: Romanian TST 2022, Day 3 P3

Tags: number theory, prime numbers, congruence



Consider a prime number $p\geqslant 11$. We call a triple $a,b,c$ of natural numbers suitable if they give non-zero, pairwise distinct residues modulo $p{}$. Further, for any natural numbers $a,b,c,k$ we define \[f_k(a,b,c)=a(b-c)^{p-k}+b(c-a)^{p-k}+c(a-b)^{p-k}.\]Prove that there exist suitable $a,b,c$ for which $p\mid f_2(a,b,c)$. Furthermore, for each such triple, prove that there exists $k\geqslant 3$ for which $p\nmid f_k(a,b,c)$ and determine the minimal $k{}$ with this property. Călin Popescu and Marian Andronache