A minimum problem for finite sets of real numbers with nonnegative sum

Let n and r be two integers such that 0 < r < n; we denote by γ(n,r)[η(n,r)] the minimum [maximum] number of the nonnegative partial sums of a sum ∑ 1=1 n a i ≥ 0,where a 1,...,a n are n real numbers arbitrarily chosen in such a way that r of them are nonnegative and the remaining n - r are negative. we study the following two problems: (p 1) which are the values of γ(n,r) and η(n,r) for each n and r,0 < r ≤ n? (p2) if q is an integer such that γ(n,r) ≤ q ≤ η(n,r),can we find n real numbers a 1,...,a n,such that r of them are nonnegative and the remaining n - r are negative with ∑ 1=1 n a i ≥ 0,such that the number of the nonnegative sums formed from these numbers is exactly q? © 2012 g. chiaselotti et al.