For any nonempty set S of real numbers, let σ(S) denote the sum of the elements of S. Given a set A of n positive integers, consider the collection of all distinct sums σ(S) as 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. ratioinequalitiesnumber theory unsolvednumber theory