101. Is it possible to plan a city's bus network, consisting of 1310 routes, in such a way that after the closure of any of these routes, it would still be possible to travel from any existing bus stop to any other (possibly with transfers along the way), but after the closure of any two routes, there would inevitably be stops from which it would no longer be possible to travel to another?
Problem 1031
Official solution
101. Possibly. Consider, for example, 10 lines on a plane such that no two of them are parallel and no three intersect at the same point. We will consider these lines as bus routes, and their points of intersection as stops. In this case, from each stop, one can travel to any other: if the stops lie on the same line, then without transferring, and if not, then with one transfer. Furthermore, even if one of these lines is removed from this scheme, it will still be possible to travel from each stop to any other, making no more than one transfer along the way. However, if two lines are removed, one stop - the point of intersection of these lines - will no longer be served by the remaining routes, and it will be impossible to travel from this stop to any other.
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