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2024 Rioplatense Mathematical Olympiad, 3
Tags: number theory , tn , rioplatense
Let $a$, $b$, $c$ be positive integers. Prove that for infinitely many positive odd integers $n$, there exists an integer $m > n$ such that $a^n + b^n + c^n$ divides $a^m + b^m + c^m$.