RO-TST 2008 Problem 16

Let $n \geq 3$ be a positive integer and let $m \geq 2^{n-1}+1$ . Prove that for each family of nonzero distinct subsets $(A_j)_{j \in \overline{1, m}}$ of $\{1, 2, ..., n\}$ there exist $i$ , $j$ , $k$ such that $A_i \cup A_j = A_k$ .