ILL 1986 Problem 13

Let $N = \{1, 2, \ldots, n\}$ , $n \geq 3$ . To each pair $i \neq j$ of elements of $N$ there is assigned a number $f_{ij} \in \{0, 1\}$ such that $f_{ij} + f_{ji} = 1$ . Let $r(i)=\sum_{i \neq j} f_{ij}$ , and write $M = \max_{i\in N} r(i)$ , $m = \min_{i\in N} r(i)$ . Prove that for any $w \in N$ with $r(w) = m$ there exist $u, v \in N$ such that $r(u) = M$ and $f_{uv}f_{vw} = 1$ .