8. (9) In the country, there are 101 airports. Each possible route is served by exactly one airline (in both directions). It is known that no airline can organize a round trip that includes more than two cities without repeating any city. What is the minimum possible number of airlines?
8. Answer: 51 companies.

Note that if there are no more than 50 companies, then one of them contains at least 101 flights, in which case there is a cyclic route. Example. We can assume that the airports are located at the vertices of a regular 101-gon and one of them is the main one. Let one of the companies serve all flights from the main airport, and each of the others serve flights between non-main airports at a given distance (its own for each airline).