MathDB

For any nonempty set

S

of real numbers, let

\sigma(S)

denote the sum of the elements of

S

. Given a set

A

n

positive integers, consider the collection of all distinct sums

\sigma(S)

S

ranges over the nonempty subsets of

A

. Prove that this collection of sums can be partitioned into

n

classes so that in each class, the ratio of the largest sum to the smallest sum does not exceed 2.

Problem Statement