Subcontests
(8)pmo problem 6
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. pmo problem 2
Let n be a positive integer. Show that there exists a one-to-one function σ:{1,2,...,n}→{1,2,...,n} such that
k=1∑n(k+σ(k))2k<21.