Given an integer $\mathrm{n} \geq 3$, let $\mathrm{A}_{1}, \mathrm{~A}_{2}, \ldots, \mathrm{~A}_{2 \mathrm{n}}$ be pairwise distinct non-empty subsets of the set $\{1,2, \ldots, \mathrm{n}\}$, and let $A_{2 n+1}=A_{1}$. Find the maximum value of $\sum_{i=1}^{2 n} \frac{\left|A_{i} \cap A_{i+1}\right|}{\left|A_{i}\right| \cdot\left|A_{i+1}\right|}$.
Question 226, Solution: We first prove $\frac{\left|A_{1} \cap A_{1+1}\right|}{\left|A_{i}\right| \cdot\left|A_{i+1}\right|} \leq \frac{1}{2}(i=1, 2, 3, \ldots, 2 n)$.
If max $\left\{\left|A_{i}\right|,\left|A_{i+1}\right|\right\}=1$, then $\left|A_{i} \cap A_{i+1}\right|=0$, thus $\frac{\left|A_{1} \cap A_{i+1}\right|}{\left|A_{i}\right| \cdot\left|A_{i+1}\right|}=0<\frac{1}{2}$; on the other hand, let $A_{1}=\{1\}, A_{2}=\{1,2\}, A_{3}=\{2\}, A_{4}=\{2,3\}, A_{5}=\{3\}, A_{6}=$

In summary, the maximum value of $\sum_{i=1}^{2 n} \frac{\left|A_{1} \cap A_{i+1}\right|}{\left|A_{1}\right| \cdot \left|A_{i+1}\right|}$ is $n$.