Solution of a problem on non-negative subset sums

Let n and r be positive integers with 1 ≤ 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 q with 2 n − 1 + 1 ≤ q ≤ 2 n − 2 n − r + 1 there exists an n -tuple ( a 1 , … , a n ) of integers such that ∑ i = 1 n a i ≥ 0 , a 1 , … , a r ≥ 0 , a r + 1 , … , a n < 0 and there are exactly q subsets X of { 1 , … , n } with ∑ i ∈ X a i ≥ 0 .