2010 AIME II Problems/Problem 2
Contents
Problem 2
A point is chosen at random in the interior of a unit square . Let denote the distance from to the closest side of . The probability that is equal to , where and are relatively prime positive integers. Find .
Solution
Any point outside the square with side length that has the same center and orientation as the unit square and inside the square with side length that has the same center and orientation as the unit square has .
Since the area of the unit square is , the probability of a point with is the area of the shaded region, which is the difference of the area of two squares.
Thus, the answer is
Solution
First, let's figure out which isThen, is a square inside , soTherefore, the probability that isSo, the answer is
See also
2010 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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