$8n$ players $P_1, P_2, \ldots, P_{8n}$ play a knock out tournament. It is known that all the players are of equal strength. The tournament is held in 3 rounds where the players are paired at random in each round. If it is given that $P_1$ wins in the third round. The probability that $P_2$ looses in the second round is:
Step-by-Step Solution
Key Concept: Three points lying on a circle imposes linear constraints on the circle's parameters, and the consistency of this system determines whether rational points can exist.
Given circle $x^2 + y^2 + 2gx + 2fy + c = 0$ passes through three rational points $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$, we obtain three linear equations in the parameters $g$, $f$, $c$: $x_i^2 + y_i^2 + 2gx_i + 2fy_i + c = 0$ for $i=1,2,3$. This homogeneous system of three equations in three unknowns has a non-trivial solution only if the coefficient determinant equals zero, which provides a constraint on the existence of such rational points on the circle.
Correct Answer: 4