APMO 2026 Problem 2

Let $n \geq 2$ be an integer. Let $A_1, A_2, \cdots, A_{2^n}$ be the $2^n$ subsets of an $n$-element set in some order. Prove that

$$|A_1 \cap A_2| + |A_2 \cap A_3| + \cdots + |A_{2^n-1} \cap A_{2^n}| + |A_{2^n} \cap A_1| \geq 2^{n-2}.$$