RIOPLATENSE 2018 Problem 3

Determine all the triples $\{a, b, c \}$ of positive integers coprime (not necessarily pairwise prime) such that $a + b + c$ simultaneously divides the three numbers $a^{12} + b^{12}+ c^{12}$ , $a^{23} + b^{23} + c^{23}$ and $a^{11004} + b^{11004} + c^{11004}$