Problem 8.2. In a volleyball tournament for the Euro- African cup the European teams are 9 more than the African teams. Every two teams met exactly once and the European teams gained 9 times more points than the African teams (the winner takes 1 point and the loser takes 0 point). What are the maximum possible points gained by an African team?

Ivan Tonov
8.2. Denote by $x$ the number of African teams. Then the number of European teams equals $x+9$. The African teams played each other $\frac{(x-1) x}{2}$ games and therefore the points won by them are $\frac{(x-1) x}{2}+k$, where $k$ is the number of wins over European teams.

Further, the points won by the Europeans are $\frac{(x+8)(x+9)}{2}+x(x+9)-k$. Thus,

$$
9\left(\frac{(x-1) x}{2}+k\right)=\frac{(x+8)(x+9)}{2}+x(x+9)-k
$$

and so $3 x^{2}-22 x+10 k-36=0$. Since $x$ is a positive integer, we have that $121-3(10 k-36)=229-30 k$ is a perfect square. Then $k \leq 7$ and a direct verification shows that we obtain perfect squares only for $k=2$ and $k=6$. For $k=2$ we have $x=8$ and therefore the best African team could have at most $7+2=9$ points.

For $k=6$ we get $x=6$ and therefore there are 6 African and 15 European teams. In this case the best African team has at most $5+6=11$ points, which happens if it wins over all other African teams and 6 European teams (the
other African teams lost their games against all European teams). Finally, the answer is 11 .