Two women and some men participated in a chess tournament in which every participant played two games with each of the other participants. If the number of games that the men played between themselves exceeds the number of games that the men played with the women by 66, then the number of men who participated in the tournament lies in the interval : [8,9], [10,12], (11,13], (14,17)

Let there be 2 women and x men.
Total number of participants = x + 2
Number of games played by each participant = 2(x+1)
Total number of games played = (x+2) * 2(x+1) = (x+2)(2x+2)
Number of games played between men = x * (x-1) * 2 /2 = x(x-1)
Number of games played between men and women = 2 * x * 2 = 4x
Given that the number of games that the men played between themselves exceeds the number of games that the men played with the women by 66.
x(x-1) - 4x = 66
x² - x - 4x -66 = 0
x² - 5x - 66 = 0
(x-11)(x+6) = 0
x = 11 or x = -6
Since x represents the number of men, it cannot be negative.
Therefore x = 11
The number of men who participated in the tournament is 11. The interval (11,13] contains 11.