The maximum number of balancing sets (Q1092046)

Let \(a_ 1,...,a_ n\) be a sequence of nonzero real numbers such that \(\sum^{n}_{i=1}a_ i=0\). B is called a balancing set if \(\sum_{b\in B}a_ b=0\). Let f(n) be the maximum number of balancing sets. It is shown that \(f(n)=\left( \begin{matrix} 2k\\ k\end{matrix} \right)\) if \(n=2k\) and \(f(n)=2\left( \begin{matrix} 2k\\ k-1\end{matrix} \right)\) if \(n=2k+1\).