P.M.

In the country, there are 15 cities, some of which are connected by air routes belonging to three airlines. It is known that even if any one of the airlines ceases operations, it will still be possible to travel from any city to any other (possibly with layovers), using flights of the remaining two airlines. What is the minimum number of air routes that can exist in the country?
Let the number of routes of the airlines be denoted by $a, b$, and $c$. If we close the last $c$ air routes, the graph will remain connected, so $a+b \geq 14$ (see problem $\underline{31098}$ g). Similarly, $b+c \geq 14, c+a \geq 14$. Adding these inequalities, we get: $2(a+b+c) \geq 42$, which means the three companies together have at least 21 air routes.

It remains to provide an example with 21 air routes. Any of the examples shown in the figures will do.
![](https://cdn.mathpix.com/cropped/2024_05_06_0e910282974b56b80e3ag-41.jpg?height=486&width=1438&top_left_y=2440&top_left_x=302)