# Minimum number of balanced partitions

For any multiset $$x_1,x_2,\ldots,x_{2n}$$ of positive real numbers, a partition into two nonempty subsets $$(A,B)$$ is called "balanced" if $$\text{sum}(A)\geq\text{sum}(B)-\max(B)$$ and $$\text{sum}(B)\geq\text{sum}(A)-\max(A)$$.

What is the minimum number of balanced partitions, in terms of $$n$$?

• I asked a question on math.SE that I believe would lead to a decent lower bound for this problem (at least $\tfrac12\binom{n}{n/2}$ I think): Reverse Littlewood-Offord problem – Dap Jan 30 at 8:03