CMO 2023 Problem 6

There are $n(n\ge 8)$ airports, some of which have one-way direct routes between them. For any two airports $a$ and $b$ , there is at most one one-way direct route from $a$ to $b$ (there may be both one-way direct routes from $a$ to $b$ and from $b$ to $a$ ). For any set $A$ composed of airports $(1\le | A| \le n-1)$ , there are at least $4\cdot \min \{|A|,n-|A| \}$ one-way direct routes from the airport in $A$ to the airport not in $A$ . Prove that: For any airport $x$ , we can start from $x$ and return to the airport by no more than $\sqrt{2n}$ one-way direct routes.