A probability problem by Pratik Vora
The football league of a certain country is played according to the following rules:
1) Each team plays exactly one game against each of the other teams.
2) The winning team of each game is awarded 1 point and the losing team gets 0 points.
3) If a match ends in a draw, both the teams get half a point.
After the league was over, the teams were ranked according to the points that they had earned at the end of the tournament. Analysis of the points table revealed the following:
a) Exactly half of the points earned by each team were earned in games against the ten teams which finished at the bottom of the table.
b) 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?
The answer is 25.
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4 solutions
It is also intersting to change "the ten players with the least number of points" to "the m players with the least number of points."
If we again let the total number of teams be n + m , then we obtain n + m = ( n − m ) 2 .
This problem was copied from another source, though I don't remember where I saw it before, but I remember solving this problem.
Let's say there are n + 1 0 teams, where n represents the teams that aren't in the bottom 10.
The bottom 10 teams won ( 2 1 0 ) = 4 5 games/points against each other, so they also won 45 games/points against the top n teams. Following the same reasoning, the top n teams won ( 2 n ) = 2 n ( n − 1 ) games/points against each other and against the bottom 10 teams
There are 10n games/points exchanged between the top n teams and the bottom 10 teams. This is also equal to 4 5 + 2 n ( n − 1 ) .
4 5 + 2 n ( n − 1 ) = 1 0 n
9 0 + n ( n − 1 ) = 2 0 n
n 2 − 2 1 n + 9 0 = 0
( n − 6 ) ( n − 1 5 ) = 0
If n = 6, those n teams wouldn't be the top n teams, so n must equal 15. So, 1 5 + 1 0 = 2 5
There are 10 teams in the bottom group and say n teams in the top group. The bottom group gets 45 points (there are 45 matches and 1 point per match) 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.
nC2 = 10n – 45
n(n + 1) = 20n – 90
n2 – 21n + 90 = 0
(n – 6) (n – 15) = 0
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 or an average of 9. This is not possible. Therefore, n = 15
The total number of teams is 10 + 15 or 25.
There are 10 teams in the bottom group and say n teams in the top group. The bottom group gets 45 points (there are 45 matches and 1 point per match) 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.
nC2 = 10n – 45 n(n + 1) = 20n – 90 n2 – 21n + 90 = 0 (n – 6) (n – 15) = 0
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 or an average of 9.
This is not possible. n = 15
The total number of teams is 10 + 15 or 25.
See 1985 AIME #14 .