Question

The football league of a certain country is played according to the following rules:
- Each team plays exactly one game against each of the other teams.
- The winning team of each game is awarded 1 point and the losing team gets 0 point.
- If a match ends in a draw, both the teams get \(\tfrac{1}{2}\) point.
At the end of the tournament, it was found that:
- Exactly half of the points earned by each team were in games against the ten teams which finished at the bottom of the table.
- Each of the bottom ten teams earned half of their total points against the other nine teams in the bottom ten.
How many teams participated in the league?

• A. 16
• B. 18
• C. 19
• D. 25
• E. 30

Hint:

In such league problems, always split the group into top and bottom and use symmetry conditions (like “half points” rules) to form equations.

Answer 1

The Correct Option is D

Step 1: Define variables.
- Number of teams in bottom group = 10. - Let number of teams in top group = \(n\). - Total number of teams = \(10 + n\).

Step 2: Points within bottom group.
Matches among bottom 10 teams = \(\binom{10}{2} = 45\). Each match gives 1 point total. So, total points among bottom group = 45. Since each bottom team earns half its points from within bottom group: Total bottom points = \(45 \times 2 = 90\). So, bottom teams earned 45 from within and 45 from matches vs. top teams.

Step 3: Points for top vs bottom.
Total points scored in matches between top and bottom groups = 45. Therefore, total points from these matches = \(10n - 45\).

Step 4: Points within top group.
Points from matches among top group = \(\binom{n}{2} = \frac{n(n-1)}{2}\). So, total top points = \(\frac{n(n-1)}{2} + (10n - 45)\).

Step 5: Apply given condition.
Each top team earned half its points from bottom 10, half from top group. So: \[ \frac{n(n-1)}{2} = 10n - 45 \]

Step 6: Solve equation.
\[ n^2 - n = 20n - 90 \] \[ n^2 - 21n + 90 = 0 \] \[ n = 15 \quad \text{or} \quad n = 6 \]

Step 7: Check feasibility.
- If \(n=6\): average top team points \(= \frac{\binom{6}{2}+60-45}{6} = 5\). - Bottom average points = 9. This is not possible since bottom cannot outscore top group. - If \(n=15\): valid.

Step 8: Final result.
\[ \text{Total teams} = 10 + 15 = 25 \]

Final Answer:
\[ \boxed{25} \]