Question 2
Let $n\ge2$ be an integer. Let $A_1,A_2,\ldots,A_{2^n}$ be all $2^n$ subsets of an $n$-element set, in some order. Prove that
$$|A_1\cap A_2|+|A_2\cap A_3|+\cdots+|A_{2^n-1}\cap A_{2^n}|+|A_{2^n}\cap A_1|\ge2^{n-2}.$$
$$|A_1\cap A_2|+|A_2\cap A_3|+\cdots+|A_{2^n-1}\cap A_{2^n}|+|A_{2^n}\cap A_1|\ge2^{n-2}.$$