Question 166
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)
(Anton Trygub)