Question 170
There are some direct one-way flights among $n \ge 8$ airports. Between any two airports $a$ and $b$, there is at most one direct one-way flight from $a$ to $b$ (it is possible to have direct one-way flights both from $a$ to $b$ and from $b$ to $a$). Suppose that, for any set $A$ consisting of some airports with $1 \le |A| \le n-1$, there are at least $4 \cdot \min\{|A|, n-|A|\}$ flights in total departing from the airports in $A$ and arriving at the airports not in $A$.
Prove that for any airport $x$, one can depart from $x$ and take no more than $\sqrt{2n}$ flights to return to $x$.
Prove that for any airport $x$, one can depart from $x$ and take no more than $\sqrt{2n}$ flights to return to $x$.