\section*{Problem 4 - 081014}

Some students from grades 9 and 10 of a school participated in a chess tournament. Each participant played exactly one game with every other participant. A win earned one point, a draw earned half a point. Although exactly 10 times as many students from grade 10 as from grade 9 participated, they only scored \(4 \frac{1}{2}\) times as many points as the students from grade 9.

How many participants from grade 9 were there, and how many points did they score?
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Preamble: \(n, m \in \mathbb{N} ; P_{n}\) and \(P_{m}\) are positive fractional numbers

Class 9: \(m\) students, \(P_{m}\) points

Class 10: \(n=10 \mathrm{~m}\) students, \(P_{n}=4.5 \cdot P_{m}\) points

The total number of games is equal to the sum of all points, because in each game 1 point is awarded to the winner and none to the loser (total: 1 point) or 0.5 points each (total: 1 point):

\[
\begin{aligned}
\frac{(n+m)(n+m-1)}{2} & =P_{n}+P_{m} \\
\frac{(10 m+m)(10 m+m-1)}{2} & =4.5 P_{m}+P_{m}=5.5 P_{m} \\
11 m \cdot(11 m-1) & =11 P_{m} \\
P_{m} & =m \cdot(11 m-1)
\end{aligned}
\]

The maximum possible points for the students of Class 9 is (points from games among the students of Class 9 + all games between students of Class 9 and 10 were won):

\[
\left(P_{m}\right)_{\max }=0.5 \cdot m \cdot(m-1)+m \cdot n=0.5 \cdot m \cdot(m-1)+10 m^{2}
\]

Obviously, the maximum possible points should be greater than or equal to the actual points achieved:

\[
\begin{aligned}
P_{m} & =m(11 m-1) \leq 0.5 \cdot m \cdot(m-1)+10 m^{2} \\
11 m^{2}-m & \leq 0.5 m^{2}-0.5 m+10 m^{2} \\
22 m^{2}-2 m & \leq 21 m^{2}-m \\
m^{2}-m=m \cdot(m-1) & \leq 0
\end{aligned}
\]

This inequality is only satisfied for natural \(m\) when one of the factors is zero, i.e., at \(m=0\) and \(m=1\), since it can never be true simultaneously that \(m<0\) or \(m>0\) and \(m-1<0\).

Obviously, \(m=0\) is not a meaningful solution, as it would correspond to a chess tournament with no participants...

Therefore: One student from Class 9 participated \((m=1)\). He achieved 10 points (according to equation (1) \(P_{m}=10\) )