Question 87
Assume that $n$ is a positive integer, and $A_1, A_2, \dots, A_{n+1}$ are $n+1$ nonempty subsets of the set $\{1, 2, \dots, n\}$. Prove that there are two disjoint and nonempty subsets $\{i_1, i_2, \dots, i_k\}$ and $\{j_1, j_2, \dots, j_m\}$ such that
$$
A_{i_1} \cup A_{i_2} \cup \dots \cup A_{i_k} = A_{j_1} \cup A_{j_2} \cup \dots \cup A_{j_m}.
$$
$$
A_{i_1} \cup A_{i_2} \cup \dots \cup A_{i_k} = A_{j_1} \cup A_{j_2} \cup \dots \cup A_{j_m}.
$$