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a+b+c divides \Sigma a^{12}, \Sigma a^{23}, \Sigma a^{11004}, coprime

Source: Rioplatense Olympiad 2018 level 3 p3

December 11, 2018

number theorycoprimecoprime numbersSum of powersdivides

Problem Statement

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}