## In the city, there are 57 bus routes. It is known that:

1) from any stop, you can get to any other stop without transferring;
2) for each pair of routes, there is one, and only one, stop where you can transfer from one of these routes to the other
3) each route has no fewer than three stops.

How many stops does each of the 57 routes have?
Let there be $n$ stops on some route $a$. Take a stop $B$ that route $a$ does not pass through. From $B$, there is a route to each of the $n$ stops on route $a$, and there is exactly one such route, since two different routes cannot have two common stops. Each route passing through $B$ intersects route $a$. Therefore, exactly $n$ routes pass through $B$.

Now it is clear that any route $b$ that does not pass through stop $B$ has exactly $n$ stops.

Indeed, as above, the number of stops on it is equal to the number of routes passing through $B$. Consider an arbitrary point $A$ on route $a$, take a stop $C$ on route $A B$ different from $A$ and $B$, and draw a route $c$ through it, different from $A B$. Route $c$ does not pass through $B$, so it has $n$ stops. Stop $A$ does not lie on $c$, so, as shown above, $n$ routes (including $a$) pass through $A$. The same is true for each point on route $a$.

Thus, a total of $n(n-1)$ routes, different from $a$, pass through the points of route $a$, and there is also the route $a$ itself. These are all the routes in the city.

Solving the equation $n(n-1)+1=57$, we find that $n=8$.

## Answer

8 stops.