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The answer is: n=3

Here are two typical solutions:



There in all 3n(3n-1)/2 matches.  These can be broken down into M-M,
M-F, an d F-F.  The number of M-M matches is n(2n-1).  Since all of
these must be won by men,

n(2n-1) <= (5/12) (9n^2 -3n)/2, which solves to  n<= 3.

We are then left with three possible scenarios:
6 men; 3 women
4 men; 2 women
2 men; 1 woman

One can begin with any of these, but it can be found that the first
works in the case that women beat men in all mixed matches.  The
others do not have enough matches for there to be a 7/5 ratio (i.e.
the number of matches is not divisible by 12).




The total number of matches is 3n(3n-1)/2.  Of these, 5n(3n-1)/8 (5/12
of the total) were won by men.  Since this number must be an integer,
n = 0 or 3 (mod 8).

There were n(2n-1) matches between men, and of course all of these
matches were won by men.  Therefore, the number of mixed matches won
by men is 5n(3n-1)/8 - n(2n-1) = (-n^2 + 3n)/8.

The total number of mixed matches is 2n^2, so 0 <= (-n^2 + 3n)/8 <=
2n^2.  The right side of the inequality will hold for any n >= 1, and
the left side will hold for n <= 3.

CONCLUSION: There were 3 women and 6 men.  All of the mixed matches
were won by women.