Question 175
Suppose there are $2010$ airports. Each airport has a number of direct flights to some of the other airports and the following conditions (1), (2) are known to be satisfied:
(1) For any pair of airports, say $A$ and $B$, one can go from $A$ to $B$, by making connections of several direct flights.
(2) If any one of the direct flights currently in operation is canceled, then the condition (1) will no longer be valid.
One day one of the direct flights in operation is canceled. How many possible ways are there for opening a new direct flight (which may be the same as the canceled one) in order to ensure that both of the conditions (1) and (2) above will be satisfied?
Note that even when there is a direct flight from airport $X$ to airport $Y$ it is not necessarily true that there is a direct flight from $Y$ to $X$.
(1) For any pair of airports, say $A$ and $B$, one can go from $A$ to $B$, by making connections of several direct flights.
(2) If any one of the direct flights currently in operation is canceled, then the condition (1) will no longer be valid.
One day one of the direct flights in operation is canceled. How many possible ways are there for opening a new direct flight (which may be the same as the canceled one) in order to ensure that both of the conditions (1) and (2) above will be satisfied?
Note that even when there is a direct flight from airport $X$ to airport $Y$ it is not necessarily true that there is a direct flight from $Y$ to $X$.