MathDB

N1

Part of 2021 Romanian Master of Mathematics Shortlist

Problems(1)

Divisors and Unit Fractions

Source: RMM Extralist 2021 N1

9/18/2023
Given a positive integer NN, determine all positive integers nn, satisfying the following condition: for any list d1,d2,…,dkd_1,d_2,\ldots,d_k of (not necessarily distinct) divisors of nn such that 1d1+1d2+…+1dk>N\frac{1}{d_1} + \frac{1}{d_2} + \ldots + \frac{1}{d_k} > N, some of the fractions 1d1,1d2,…,1dk\frac{1}{d_1}, \frac{1}{d_2}, \ldots, \frac{1}{d_k} add up to exactly NN.
RMM Shortlistnumber theoryunit fractionsDivisors