2006 AIME I Problems/Problem 10
Problem
Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a 
chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The total points are accumilated to decide the ranks of the teams. In the first game of the tournament, team 
beats team 
The probability that team finishes with more points than team 
is 
where 
and 
are relatively prime positive integers. Find 
Solution
Assume that if unit squares are drawn circumscribing the circles, then the line will divide the area of the concave hexagonal region of the squares equally (proof needed). Denote the intersection of the line and the x-axis as 
.
The line divides the region into 2 sections. The left piece is a trapezoid, with its area 
. The right piece is the addition of a trapezoid and a rectangle, and the areas are 
and 
, totaling 
. Since we want the two regions to be equal, we find that 
, so 
.
We have that 
is a point on the line of slope 3, so 
and 
. In y-intercept form, the equation of the line is 
, and in the form for the answer, the line’s equation is 
. Thus, our answer is 
.
See also
| 2006 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 9 |
Followed by Problem 11 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
