In a chess tournament, only grandmasters and masters participated. The number of the latter was 3 times that of the grandmasters, and their total points achieved were 1.2 times the total points of the grandmasters.

How many people participated in the tournament? What can we say about the outcome of the tournament?
a) Let the number of grandmasters participating be $x$, so the number of masters is $3x$, the total number of participants is $4x$, and the number of matches played, and thus the total number of points scored, is $4x(4x-1)$, since each participant played every other participant twice, once with white pieces and once with black pieces. - According to the other data, the masters collectively scored $1.2(1+1.2)=6/11$ of the total points.

On the other hand, the masters collectively scored at least as many points as the number of matches where both players were masters, which is similarly $3x(3x-1)$ points. Therefore,

$$
\frac{6}{11} 4x(4x-1) \geq 3x(3x-1)
$$

Multiplying by 11 and dividing by $3x$,

$$
x \leq 3
$$

However, $x$ cannot be 1 or 2, because in these cases the left side of (1) is not an integer, nor is it a number greater than an integer by 0.5. With $x=3$, however, it is an integer, so this is the (only) solution: 3 grandmasters and 9 masters participated in the tournament.

b) According to the solution found, equality holds in (1), meaning the masters collectively have exactly as many points as the number of matches they played among themselves, so no master ever scored a point against a grandmaster. Thus, no master can have more than 16 points, which is the total if each master defeated every other master twice. On the other hand, each grandmaster scored at least 18 points by defeating each of the 9 masters twice. Therefore, the top 3 places in the tournament were the 3 grandmasters (possibly in a tie).

Gáspár Gyula (Miskolc, Herman O. Gimn., II. o. t.) Orosz Éva (Székesfehérvár, Teleki Blanka Gimn., II. o. t.)

Remarks. 1. The above approach can also be completed as follows. For the grandmasters collectively, the number of points they can score in the $x(x-1)$ matches among themselves is certain. In the additional $x \cdot 3x \cdot 2 = 6x^2$ matches, each was played by one master and one grandmaster. Let the total number of points scored by the masters in these matches be $m$, so the remaining $6x^2 - m$ points went to the grandmasters and

$$
\begin{gathered}
3x(3x-1) + m = 1.2 \{x(x-1) + 6x^2 - m\} \\
3x^2 - 9x + 11m = 0 \\
x = \frac{1}{6} \{9 \pm \sqrt{81 - 132m}\}
\end{gathered}
$$

$x$ is only real for $m \leq 27/44$, but for $m=0.5$, $x$ is not rational. For $m=0$, however, $x=3$.

Árvay László (Budapest, Kölcsey F. Gimn., II. o. t.)

2. Of course, we arrive at the same results from the assumption that any two players played each other only once.

[^0]:    ${ }^{1}$ In chess, the winner of each match scores 1 point, or in the case of a draw, each player scores $0.5-0.5$ points.