Can someone check this solution: Let the number of teams from Africa be $n$, so the number of teams from Europe is $n+9$ meaning $2n+9$ teams were there overall. Therefore the total amount of games were ${2n+9 \choose 2}$. Thus the total amount of wins the African teams had in total is ${2n+9 \choose 2}/9$. To maximize a single African teams wins we give the rest of the African teams 0 wins and one team ${2n+9 \choose 2}/9$ wins.

mathlete5451006 wrote: Can someone check this solution: Let the number of teams from Africa be $n$, so the number of teams from Europe is $n+9$ meaning $2n+9$ teams were there overall. Therefore the total amount of games were ${2n+9 \choose 2}$. Thus the total amount of wins the African teams had in total is ${2n+9 \choose 2}/9$. To maximize a single African teams wins we give the rest of the African teams 0 wins and one team ${2n+9 \choose 2}/9$ wins. It should be $\frac{\binom{2n+9}{2}}{10}$. Also this is only valid when $n < 6$ because: The maximum games won by a single African team is $(n+9) + (n-1)$ because it can only win $n+9$ games max with other European teams and $n-1$ with other African teams. So $\frac{\binom{2n+9}{2}}{10} \leq n+9+n-1=2n+8 \implies 2n+9 \leq 20 \implies n \leq 5$

huh why am i getting something different from everyone else above We claim that the answer is 11. Let $n$ be the number of African teams. Let $k$ be the number of games in which a European team defeated an African team. Then, the condition states that $${n+9\choose 2}+k=9({n\choose 2}+(n(n+9)-k)).$$After expanding everything, this becomes $$13n^2+68n=10k+36.$$Clearly, $k\leq n(n+9),$ so $$10k+36\leq 10n^2+90n+36.$$Thus, $$13n^2+68n\leq 10n^2+90n+36.$$This implies that $n\leq 8$ since $n$ is an integer. Note that $$k=\frac{13n^2+68n-36}{10}.$$Testing $n$ from 1 to 8 gives that only $n=6$ and $n=8$ give integer $k$, which give $k=84$ and $k=134$ respectively. If $n=6$ and $k=84$, there are 6 African teams and 15 European teams, and there are 6 games in which an African team beat a European team. To maximize the number of wins of one African team, have all 6 of these wins be attributed to the same African team. Since they have can have up to 5 more wins (from other African teams), the maximum possible is 11. In the other case of $n=8$ and $k=134$, the maximum is $7+2=9$, which is not as high. Thus, the answer is 11, achieved when: -There are 15 European teams and 6 African teams. -One of the African teams wins against all 5 other African teams. -That same African team beats 6 out of the 15 European teams. All other games between African and European teams results in a win for the European team. This clearly works since ${15\choose 2}+84=189=9({6\choose 2}+6)$ so we are done