1. Given an integer $n(n \geqslant 3)$, let $A_{1}, A_{2}, \cdots, A_{2 n}$ be pairwise distinct non-empty subsets of the set $\{1,2, \cdots, n\}$, and let $A_{2 n+1}=A_{1}$. Find
$$
\sum_{i=1}^{2 n} \frac{\left|A_{i} \cap A_{i+1}\right|}{\left|A_{i}\right| \cdot\left|A_{i+1}\right|}
$$

the maximum value.
(Liang Yingde)
1. For any $i$, if $\left|A_{i} \cap A_{i+1}\right|=0$, then
$$
\frac{\left|A_{i} \cap A_{i+1}\right|}{\left|A_{i}\right| \cdot\left|A_{i+1}\right|}=0.
$$

Assume $\left|A_{i} \cap A_{i+1}\right| \geqslant 1$ below.
Since $A_{i} \cap A_{i+1} \subseteq A_{i}$ and $A_{i} \cap A_{i+1} \subseteq A_{i+1}$, we have $\left|A_{i} \cap A_{i+1}\right| \leqslant \min \left(\left|A_{i}\right|,\left|A_{i+1}\right|\right)$.
Also, $A_{i} \neq A_{i+1}$ and $\left|A_{i} \cap A_{i+1}\right| \geqslant 1$, then $\max \left(\left|A_{i}\right|,\left|A_{i+1}\right|\right) \geqslant 2$.
Therefore, $\sum_{i=1}^{2 n} \frac{\left|A_{i} \cap A_{i+1}\right|}{\left|A_{i}\right| \cdot\left|A_{i+1}\right|}$
$$
\begin{array}{l}
\leqslant \sum_{i=1}^{2 n} \frac{1}{\max \left(\left|A_{i}\right|,\left|A_{i+1}\right|\right)} \\
\leqslant \sum_{i=1}^{2 n} \frac{1}{2}=n.
\end{array}
$$

The equality in the above formula can be achieved, for example:
$$
\begin{array}{l}
A_{1}=\{1\}, A_{2}=\{1,2\}, \cdots \cdots A_{2 i-1}=\{i\} \\
A_{2 i}=\{i, i+1\}, \cdots \cdots A_{2 n-1}=\{n\} \\
A_{2 n}=\{n, 1\}
\end{array}
$$