Problem

Source: Iran TST 2008

Tags: floor function, induction, combinatorics proposed, combinatorics, boolean lattice method



Let $ S$ be a set with $ n$ elements, and $ F$ be a family of subsets of $ S$ with $ 2^{n-1}$ elements, such that for each $ A,B,C\in F$, $ A\cap B\cap C$ is not empty. Prove that the intersection of all of the elements of $ F$ is not empty.