Solution of a problem on non-negative subset sums

Let nn and rr be positive integers with 1≤r≤n−11≤r≤n−1. Solving a problem of Chiaselotti–Marino–Nardi, which is a generalization of a problem of Manickam and Miklós, we prove that for each integer qq with 2n−1+1≤q≤2n−2n−r+12n−1+1≤q≤2n−2n−r+1 there exists an nn-tuple (a1,…,an)(a1,…,an) of integers such that ∑i=1nai≥0, a1,…,ar≥0,ar+1,…,an<0a1,…,ar≥0,ar+1,…,an<0 and there are exactly qq subsets XX of {1,…,n}{1,…,n} with ∑i∈Xai≥0∑i∈Xai≥0.