1. **Problem statement:** We have a football league where each team plays every other team exactly once. Points are awarded as follows: 3 for a win, 0 for a loss, and 1 for a draw. 2. The league is divided into two groups: the 10 weakest teams and the rest (strong teams). Each team earned exactly 75% of its total points from games against the 10 weakest teams. 3. Among all games, 60% ended in a win and 40% in a draw. This ratio holds for games between all teams, between strong teams, and between the weakest teams. 4. Let the total number of teams be $N$. The 10 weakest teams form one group, and the strong teams form the other group with $N-10$ teams. 5. Total games played in the league: $$\frac{N(N-1)}{2}$$ 6. Total games among the 10 weakest teams: $$\frac{10 \times 9}{2} = 45$$ 7. Total games among the strong teams: $$\frac{(N-10)(N-11)}{2}$$ 8. Total games between strong and weak teams: $$10 \times (N-10)$$ 9. Points per game depend on result: win = 3 points to winner, 0 to loser; draw = 1 point each, total 2 points per game. 10. Since 60% of games end in wins and 40% in draws, average points per game is: $$0.6 \times 3 + 0.4 \times 2 = 1.8 + 0.8 = 2.6$$ 11. Total points in the league: $$\text{Total games} \times 2.6 = \frac{N(N-1)}{2} \times 2.6 = 1.3 N (N-1)$$ 12. Points in games among the 10 weakest teams: $$45 \times 2.6 = 117$$ 13. Points in games among strong teams: $$\frac{(N-10)(N-11)}{2} \times 2.6 = 1.3 (N-10)(N-11)$$ 14. Points in games between strong and weak teams: $$10 (N-10) \times 2.6 = 26 (N-10)$$ 15. Each weak team earns 75% of its points from games against the other 9 weak teams. Since there are 10 weak teams, total points earned by weak teams in their internal games is 75% of their total points. 16. Let $P_w$ be total points earned by weak teams, $P_s$ by strong teams. Then: $$P_w = \text{points weak teams earn in weak-weak games} + \text{points weak teams earn in weak-strong games}$$ 17. From the problem, weak teams earn 75% of their points in weak-weak games, so: $$\text{points weak teams in weak-weak games} = 0.75 P_w$$ 18. Since total points in weak-weak games is 117, and weak teams earn all these points (because both teams are weak), $$0.75 P_w = 117 \implies P_w = \frac{117}{0.75} = 156$$ 19. Similarly, strong teams earn 75% of their points in games against the 10 weakest teams, and 25% in games against other strong teams. 20. Let $P_s$ be total points of strong teams. Points strong teams earn in strong-weak games is 75% of $P_s$: $$0.75 P_s = \text{points strong teams earn in strong-weak games}$$ 21. Total points in strong-weak games is $26 (N-10)$, split between strong and weak teams. Since weak teams earn 25% of their points in strong-weak games, and strong teams earn 75% of their points in strong-weak games, the points in strong-weak games are split accordingly. 22. Points weak teams earn in strong-weak games: $$P_w^{sw} = 0.25 P_w = 0.25 \times 156 = 39$$ 23. Points strong teams earn in strong-weak games: $$P_s^{sw} = 0.75 P_s$$ 24. Since total points in strong-weak games is $26 (N-10)$, and these points are split between weak and strong teams: $$P_w^{sw} + P_s^{sw} = 26 (N-10)$$ Substitute values: $$39 + 0.75 P_s = 26 (N-10)$$ 25. Points strong teams earn in strong-strong games is 25% of $P_s$: $$P_s^{ss} = 0.25 P_s$$ 26. Total points in strong-strong games is: $$1.3 (N-10)(N-11)$$ Since strong teams earn all points in strong-strong games: $$P_s^{ss} = 1.3 (N-10)(N-11)$$ 27. Equate: $$0.25 P_s = 1.3 (N-10)(N-11) \implies P_s = 5.2 (N-10)(N-11)$$ 28. Substitute $P_s$ into equation from step 24: $$39 + 0.75 \times 5.2 (N-10)(N-11) = 26 (N-10)$$ Simplify: $$39 + 3.9 (N-10)(N-11) = 26 (N-10)$$ 29. Rearrange: $$3.9 (N-10)(N-11) - 26 (N-10) + 39 = 0$$ 30. Factor out $(N-10)$: $$(N-10)(3.9 (N-11) - 26) + 39 = 0$$ 31. Simplify inside parentheses: $$3.9 N - 42.9 - 26 = 3.9 N - 68.9$$ So: $$(N-10)(3.9 N - 68.9) + 39 = 0$$ 32. Expand: $$3.9 N^2 - 68.9 N - 39 N + 689 + 39 = 0$$ Combine terms: $$3.9 N^2 - 107.9 N + 728 = 0$$ 33. Solve quadratic equation: $$3.9 N^2 - 107.9 N + 728 = 0$$ Divide entire equation by 3.9: $$N^2 - 27.6667 N + 186.6667 = 0$$ 34. Use quadratic formula: $$N = \frac{27.6667 \pm \sqrt{27.6667^2 - 4 \times 186.6667}}{2}$$ Calculate discriminant: $$27.6667^2 = 765.4449$$ $$4 \times 186.6667 = 746.6668$$ $$\sqrt{765.4449 - 746.6668} = \sqrt{18.7781} \approx 4.334$$ 35. Calculate roots: $$N_1 = \frac{27.6667 + 4.334}{2} = \frac{32}{2} = 16$$ $$N_2 = \frac{27.6667 - 4.334}{2} = \frac{23.333}{2} = 11.6665$$ 36. Since $N$ must be an integer and greater than 10 (because there are 10 weak teams), the total number of teams is: $$\boxed{16}$$