62nd Ukrainian National Mathematical Olympiad, Third Round, First Tour

You are given a set of $n$ not necessarily distinct numbers $\{a_1,a_2,\dots ,a_n\}$, (meaning that some of them can be equal). Consider all $2^n-1$ nonempty subsets of this set, and for each such subset, find the sum of its elements. What largest number of these sums could turn out to be equal to $1$? For example, for a set $\{-1;2;2\}$ we have $7$ nonempty subsets: $\{-1\}$, $\{2\}$, $\{2\}$, $\{-1;2\}$, $\{-1;2\}$, $\{2;2\}$ and $\{-1;2;2\}$, and among them there are exactly two subsets with sum $1$. (Anton Trygub)