10. The bus network in the city of Lisse is arranged in such a way that: a) each route has three stops; b) any two routes either have no common stops at all or have only one common stop. What is the maximum number of routes that can exist in this city, given that there are only nine different stops?
10. Let's consider some stop $A$. Define how many routes can pass through it. Besides $A$, there are eight other stops in the city. On each route passing through $A_{\text {r }}$, there are two more stops. Since no two of these routes can have common stops other than $A$, a total of no more than $8: 2=4$ routes can pass through $A$.

Let's number all the stops and denote by $a_{i}$ the number of routes passing through the $i$-th stop. Since each route has three stops, we have

$$
a_{1}+a_{2}+\ldots+a_{9}=3 n
$$

where $\eta$ is the total number of routes. By the proven fact, all terms do not exceed four. Therefore,

$$
3 n \leqslant 4 \cdot 9 = 36 \text { and } n \leqslant 12
$$

In Fig. 14, a scheme is shown that satisfies the conditions of the problem and contains 12 routes (eight straight and four curved).

![](https://cdn.mathpix.com/cropped/2024_05_21_7d0959e1f193b6338b84g-29.jpg?height=587&width=566&top_left_y=235&top_left_x=251)

Fig. 14.