A Minimum Problem for Finite Sets of Real Numbers with Nonnegative Sum

Abstract

Let $n$ and $r$ be two integers such that ; we denote by $\gamma (n,r)$ $\left[\eta \right(n,r\left)\right]$ the minimum [maximum] number of the nonnegative partial sums of a sum ${\sum }_{1=1}^n{a}_i\ge 0$, where $a_1,\dots ,a_n$ 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: $\left(P1\right)$ which are the values of $\gamma (n,r)$ and $\eta (n,r)$ for each $n$ and $r$, ? $\left(P2\right)$ if $q$ is an integer such that $\gamma (n,r)\le q\le \eta (n,r)$ , can we find $n$ real numbers $a_1,\dots ,a_n$, such that $r$ of them are nonnegative and the remaining $n-r$ are negative with ${\sum }_{1=1}^n{a}_i\ge 0$, such that the number of the nonnegative sums formed from these numbers is exactly $q$ ?