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 ½ point. After the league was over, the teams were ranked according to the points that they earned at the end of the tournament. Analysis of the points table revealed the following :


• Exactly half of the points earned by each team were earned in games against the 10 teams which finished at the bottom of the table.


• Each of the bottom 10 teams earned half of their total points against the other 9 teams in the bottom 10 . How many teams participated in the league?

Let total number of teams participated in tournament is n + 10
There are 10 teams in the bottom group then n teams in the top group. It is given that the bottom group gets 45 points since we have 1 point per match therefore 45 matches playing amongst themselves. Therefore they should get 45 points from their matches against the top group i.e., 45 out of the 10n points. The top group get ⁿC₂ points from the matches among themselves. They also get ( 10n ? 45 ) points against the bottom group, which is half their total points.
Hence ⁿC₂ = 10n ? 45 ? n(n + 1) = 20n ? 90 ? n² ? 21n + 90 = 0 hence n = 6 or 15
If n = 6, the top group would get ⁿC₂ + 10n ? 45 = ⁿC₂ + 10(6) ? 45 = 30 points, or an average of 5 points per team, while the bottom group would get (45 + 45)/10 = 90/10 = 9. This is not possible. Hence n = 15. Then total number of teams is 10 + 15 = 25.