USAMO 1996 Problem 2

For any nonempty set $S$ of real numbers, let $\sigma(S)$ denote the sum of the elements of $S$ . Given a set $A$ of $n$ positive integers, consider the collection of all distinct sums $\sigma(S)$ 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.