MathDB

The set

A{}

of positive integers satisfies the following conditions:
[*]If a positive integer

n{}

belongs to

A{}

, then

2n

also belongs to

A{}

; [*]For any positive integer

n{}

there exists an element of

A{}

divisible by

n{}

; [*]There exist finite subsets of

A{}

with arbitrarily large sums of reciprocals of elements. Prove that for any positive rational number

r{}

there exists a finite subset

B\subset A

such that

\sum_{x\in B}\frac{1}{x}=r.

Problem Statement