Question 5
In a tournament with $n$ players, numbered from 1 to $n$, each pair of players meets exactly once. Each match ends with the victory of one of the two players and the defeat of the other. Let $v_{k}$ be the number of wins of player $k$ during the tournament, and $d_{k}$ his number of losses.
Prove that $\sum_{k=1}^{n} v_{k}^{2}=\sum_{k=1}^{n} d_{k}^{2}$.
Prove that $\sum_{k=1}^{n} v_{k}^{2}=\sum_{k=1}^{n} d_{k}^{2}$.