Difference between revisions of "2010 AIME II Problems/Problem 2"
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== Problem 2 == | == Problem 2 == | ||
A point <math>P</math> is chosen at random in the interior of a unit square <math>S</math>. Let <math>d(P)</math> denote the distance from <math>P</math> to the closest side of <math>S</math>. The probability that <math>\frac{1}{5}\le d(P)\le\frac{1}{3}</math> is equal to <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | A point <math>P</math> is chosen at random in the interior of a unit square <math>S</math>. Let <math>d(P)</math> denote the distance from <math>P</math> to the closest side of <math>S</math>. The probability that <math>\frac{1}{5}\le d(P)\le\frac{1}{3}</math> is equal to <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. |
Revision as of 14:46, 9 August 2018
Contents
Problem
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 have the same center as the unit square and inside the square side length that have the same center 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
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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