KMO 2022 Problem 6

$n(\geq 4)$ islands are connected by bridges to satisfy the following conditions:

• Each bridge connects only two islands and does not go through other islands.
• There is at most one bridge connecting any two different islands.
• There does not exist a list $A_1, A_2, \ldots, A_{2k}(k \geq 2)$ of distinct islands that satisfy the following: [center]For every $i=1, 2, \ldots, 2k$ , the two islands $A_i$ and $A_{i+1}$ are connected by a bridge. (Let $A_{2k+1}=A_1$ )[/center]

Prove that the number of the bridges is at most $\frac{3(n-1)}{2}$ .