TUYMAADA 2017 Problem 3

In a country every 2 cities are connected either by a direct bus route or a direct plane flight. A $clique$ is a set of cities such that every 2 cities in the set are connected by a direct flight. A $cluque$ is a set of cities such that every 2 cities in the set are connected by a direct flight, and every 2 cities in the set are connected to the same number of cities by a bus route. A $claque$ is a set of cities such that every 2 cities in the set are connected by a direct flight, and every 2 numbers of bus routes from a city in the set are different. Prove that the number of cities of any clique is at most the product of the biggest possible number of cities in a cluque and the the biggest possible number of cities in a claque.

Tuymaada 2017 Q3 Juniors