In a chess tournament, all players faced each other exactly once. In each match, a player earns 1 point for a win, $1 / 2$ for a draw, and 0 points for a loss. At the end of the tournament, a reporter summed up the scores of all the players and obtained 190 points. In this type of tournament, the winner is the one who scores the most points.

a) How many players participated in the tournament?

b) André participated in the tournament and scored 9 points. Show that, even without knowing the other scores, André did not win the tournament.
Solution

a) Let $J$ be the number of players. Each match is worth a total of 1 point, whether it's $1+0=1$ or $1 / 2+1 / 2=1$. Therefore, the total score is equal to the number of matches. Since each of the $J$ players faces each of the other $J-1$ players, we might think that the total number of games would be $J(J-1)$ matches. However, each match ends up being counted twice, so the total number of matches is $\frac{J(J-1)}{2}$. Using the number obtained by the journalist, we have

$$
\begin{aligned}
\frac{J(J-1)}{2} & =190 \\
J(J-1) & =380 \\
& =20 \cdot 19
\end{aligned}
$$

Thus, $J=20$.

b) Since the total points were 190 for 20 competitors, the average points is $\frac{190}{20}=9.5$ points. Since André is below the average points and there is always a player who has scored at least as many points as the average, we can conclude that he was not the winner of the tournament.