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Sad Number Theory

Source: 2018 USAMO 3, by Ivan Borsenco

April 18, 2018
USAMO2018 USAMO Problem 3number theoryHi

Problem Statement

For a given integer n≥2n\ge 2, let {a1,a2,…,am}\{a_1,a_2,…,a_m\} be the set of positive integers less than nn that are relatively prime to nn. Prove that if every prime that divides mm also divides nn, then a1k+a2k+⋯+amka_1^k+a_2^k + \dots + a_m^k is divisible by mm for every positive integer kk.
Proposed by Ivan Borsenco
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