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Thailand National Olympiad
2012 Thailand Mathematical Olympiad
8
8
Part of
2012 Thailand Mathematical Olympiad
Problems
(1)
no of matches in a draw / total no of matches, 4n students tournament
Source: Thailand Mathematical Olympiad 2012 p8
8/17/2020
4
n
4n
4
n
first grade students at Songkhla Primary School, including
2
n
2n
2
n
boys and
2
n
2n
2
n
girls, participate in a taekwondo tournament where every pair of students compete against each other exactly once. The tournament is scored as follows:
∙
\bullet
∙
In a match between two boys or between two girls, a win is worth
3
3
3
points, a draw
1
1
1
point, and a loss
0
0
0
points.
∙
\bullet
∙
In a math between a boy and a girl, if the boy wins, he receives
2
2
2
points, else he receives
0
0
0
points. If the girl wins, she receives
3
3
3
points, if she draws, she receives
2
2
2
points, and if she loses, she receives
0
0
0
points. After the tournament, the total score of each student is calculated. Let
P
P
P
be the number of matches ending in a draw, and let
Q
Q
Q
be the total number of matches. Suppose that the maximum total score is
4
n
−
1
4n - 1
4
n
−
1
. Find
P
/
Q
P/Q
P
/
Q
.
combinatorics