Question 94
If $A = \{a_1, a_2, \dots, a_n\} \subset \mathbb{N}$ then let denote $A^* = \{a_i + a_j \mid 1 \le i \lt j \le n\}$ (where $A^*$ is a multiset). For instance: if $A = \{1, 2, 3, 4\}$ then $A^* = \{3, 4, 5, 5, 6, 7\}$. Prove that if $A \neq B$ and $A^* = B^*$ then $|A| = |B| = 2^k$.