Question 2
For any nonempty set $S$ of real numbers, let $\sigma(S)$ denote the sum of its elements. Given a set $A$ of $n$ positive integers, consider all distinct sums $\sigma(S)$ over nonempty subsets $S\subseteq A$. Prove that these sums can be partitioned into $n$ classes so that in each class the ratio of the largest sum to the smallest sum is at most $2$.