2006 AIME A Problems/Problem 10
Problem
Eight circles of diameter 1 are packed in the first quadrant of the coordinte plane as shown. Let region be the union of the eight circular regions. Line
with slope 3, divides
into two regions of equal area. Line
's equation can be expressed in the form
where
and
are positive integers whose greatest common divisor is 1. Find
Solution
You can break this into cases based on how many rounds A wins out of the remaining 5 games.
If A wins 0 games, then B must win 0 games and the probability of this is .
If A wins 1 games, then B must win 1 or less games and the probability of this is .
If A wins 2 games, then B must win 2 or less games and the probability of this is .
If A wins 3 games, then B must win 3 or less games and the probability of this is .
If A wins 4 games, then B must win 4 or less games and the probability of this is .
If A wins 5 games, then B must win 5 or less games and the probability of this is .
Summing these 6 cases, we get , which simplifies to
, so out answer is
.