Question 23
Let $n \ge 3$ be a natural number. Ana chooses the non-zero natural numbers $a_1, a_2, \dots, a_n$, and for each non-empty subset $A \subset \{1, 2, \dots, n\}$, she computes the sum $s_A = \sum_{k \in A} a_k$, then arranges these sums in increasing order, obtaining the sequence $s_1 \le s_2 \le \dots \le s_{2^{n}-1}$.
Show that there exists a subset $B \subset \{1, 2, \dots, 2^n - 1\}$, with $2^{n-2} + 1$ elements, such that, no matter what values Ana chooses for $a_1, a_2, \dots, a_n$, these values can be determined by knowing all the values $s_i$, for $i \in B$.
Cristi Săvescu
Show that there exists a subset $B \subset \{1, 2, \dots, 2^n - 1\}$, with $2^{n-2} + 1$ elements, such that, no matter what values Ana chooses for $a_1, a_2, \dots, a_n$, these values can be determined by knowing all the values $s_i$, for $i \in B$.
Cristi Săvescu