Problem

Source: PMO 2021

Tags: combinatorics, PMO



A certain country wishes to interconnect $2021$ cities with flight routes, which are always two-way, in the following manner: • There is a way to travel between any two cities either via a direct flight or via a sequence of connecting flights. • For every pair $(A, B)$ of cities that are connected by a direct flight, there is another city $C$ such that $(A, C)$ and $(B, C)$ are connected by direct flights. Show that at least $3030$ flight routes are needed to satisfy the two requirements.