IR-MO 2004 Problem 22

Suppose that $\mathcal F$ is a family of subsets of $X$ . $A,B$ are two subsets of $X$ s.t. each element of $\mathcal{F}$ has non-empty intersection with $A, B$ . We know that no subset of $X$ with $n - 1$ elements has this property. Prove that there is a representation $A,B$ in the form $A = \{a_1,\dots,a_n\}$ and $B = \{b_1,\dots,b_n\}$ , such that for each $1\leq i\leq n$ , there is an element of $\mathcal F$ containing both $a_i, b_i$ .