3-5. The city's bus network is organized as follows: 1) from any stop to any other stop, you can get without transferring;

2) for any pair of routes, there is, and only one, stop where you can transfer from one of these routes to the other;
3) on each route, there are exactly three stops.

How many bus routes are there in the city?
Solve 5. Answer: 7. We will prove that if the given conditions are satisfied, then the number of stops $n$ and the number of routes $N$ are related by the formula $N=n(n-1)+1$. Let $a$ be one of the routes, and $B$ be a stop that route $a$ does not pass through. Each route passing through $B$ intersects route $a$. Therefore, exactly $n$ routes pass through $B$. Similarly, it can be proven that through each stop of route $a$, $n-1$ routes different from $a$ pass. In total, we get $n(n-1)$ different routes and the route $a$ itself.

## 9 - 10 grades