Question

There were two women participating with some men in a chess tournament. Each participant played two games with the other. The number of games that the men played among themselves is 66 more than the number of games the men played with the women. Then the total number of participants in the tournament is:

• A. \( 17 \)
• B. \( 13 \)
• C. \( 11 \)
• D. \( 19 \)

Hint:

In tournament problems involving games played among participants, use combination formulas for pairwise matches and set up equations based on given conditions.

Answer 1

The Correct Option is B

Step 1: Define variables
Let the number of men in the tournament be \( m \), and the number of women be \( w = 2 \). Each participant plays two games with every other participant. The total number of games played among men is: \[ \frac{m(m-1)}{2}. \] The total number of games played between men and women is: \[ 2m. \] We are given that the number of games men played among themselves is 66 more than the games played between men and women: \[ \frac{m(m-1)}{2} = 2m + 66. \] Step 2: Solve for \( m \)
Rearrange the equation: \[ \frac{m(m-1)}{2} - 2m = 66. \] Multiply by 2 to clear the fraction: \[ m(m-1) - 4m = 132. \] Rearrange: \[ m^2 - 5m - 132 = 0. \] Solve using the quadratic formula: \[ m = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-132)}}{2(1)}. \] \[ m = \frac{5 \pm \sqrt{25 + 528}}{2}. \] \[ m = \frac{5 \pm \sqrt{553}}{2}. \] Approximating \( \sqrt{553} \approx 23.5 \): \[ m = \frac{5 \pm 23.5}{2}. \] Solving for positive \( m \): \[ m = \frac{5 + 23.5}{2} = \frac{28.5}{2} = 13. \] Step 3: Compute total participants
Total participants: \[ m + w = 13 + 2 = 13. \] Step 4: Conclusion
Thus, the total number of participants is: \[ \boxed{13}. \]