Problem

Source: 2018 Brazil 3rd TST #4

Tags: algebra, combinatorics, Subsets, inequalities, boundary conditions



Given a set S of positive real numbers, let Σ(S)={xAx:AS}.be the set of all the sums of elements of non-empty subsets of S. Find the least constant L>0 with the following property: for every integer greater than 1 and every set S of n positive real numbers, it is possible partition Σ(S) into n subsets Σ1,,Σn so that the ratio between the largest and smallest element of each Σi is at most L.