Question 34
Let $A_1, A_2, \ldots, A_k$ be subsets of $\{1, 2, 3, \ldots, n\}$ such that for all $1 \le i, j \le k : A_i \cap A_j \ne \emptyset$. Prove that there are $n$ distinct positive integers $x_1, x_2, \ldots, x_n$ such that for each $1 \le j \le k$
$$
\operatorname{lcm}_{i \in A_j} \{x_i\} \gt \operatorname{lcm}_{i \notin A_j} \{x_i\}
$$
$$
\operatorname{lcm}_{i \in A_j} \{x_i\} \gt \operatorname{lcm}_{i \notin A_j} \{x_i\}
$$