Difference between revisions of "2010 AIME II Problems/Problem 2"
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− | Any point outside the square with side length <math>\frac{1}{3}</math> that | + | Any point outside the square with side length <math>\frac{1}{3}</math> that has the same center and orientation as the unit square and inside the square with side length <math>\frac{3}{5}</math> that has the same center and orientation as the unit square has <math>\frac{1}{5}\le d(P)\le\frac{1}{3}</math>. |
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− | Since the area of the unit square is <math>1</math>, the probability of a point <math>P</math> with <math>\frac{1}{5}\le d(P)\le\frac{1}{3}</math> is the area of the shaded region, which is the difference of the area of two squares | + | Since the area of the unit square is <math>1</math>, the probability of a point <math>P</math> with <math>\frac{1}{5}\le d(P)\le\frac{1}{3}</math> is the area of the shaded region, which is the difference of the area of two squares. |
<math>\left(\frac{3}{5}\right)^2-\left(\frac{1}{3}\right)^2=\frac{9}{25}-\frac{1}{9}=\frac{56}{225}</math> | <math>\left(\frac{3}{5}\right)^2-\left(\frac{1}{3}\right)^2=\frac{9}{25}-\frac{1}{9}=\frac{56}{225}</math> |
Latest revision as of 14:36, 18 March 2020
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
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.