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Nice inverse sum

Source: Brazil National Olympiad 2019 #4

November 14, 2019
number theory

Problem Statement

Prove that for every positive integer mm there exists a positive integer nmn_m such that for every positive integer n≥nmn \ge n_m, there exist positive integers a1,a2,…,ana_1, a_2, \ldots, a_n such that 1a1m+1a2m+…+1anm=1.\frac{1}{a_1^m}+\frac{1}{a_2^m}+\ldots+\frac{1}{a_n^m}=1.
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