Abstract
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 , where a1, …, an 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: (P1) 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 a1, …, an, such thatr of them are nonnegative and the remaining n − r are negative with , such that the number of the nonnegative sums formed from these numbers is exactly q?
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