3

Part of 2018 USAMO

Problems(1)

Sad Number Theory

Source: 2018 USAMO 3, by Ivan Borsenco

For a given integer

n\ge 2

\{a_1,a_2,…,a_m\}

be the set of positive integers less than

n

that are relatively prime to

n

. Prove that if every prime that divides

m

n

a_1^k+a_2^k + \dots + a_m^k

is divisible by

m

for every positive integer

k

.
Proposed by Ivan Borsenco

USAMO2018 USAMO Problem 3number theoryHi