Difference between revisions of "2020 AIME II Problems/Problem 2"
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==Problem== | ==Problem== | ||
Let <math>P</math> be a point chosen uniformly at random in the interior of the unit square with vertices at <math>(0,0), (1,0), (1,1)</math>, and <math>(0,1)</math>. The probability that the slope of the line determined by <math>P</math> and the point <math>\left(\frac58, \frac38 \right)</math> is greater than <math>\frac12</math> can be written as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | Let <math>P</math> be a point chosen uniformly at random in the interior of the unit square with vertices at <math>(0,0), (1,0), (1,1)</math>, and <math>(0,1)</math>. The probability that the slope of the line determined by <math>P</math> and the point <math>\left(\frac58, \frac38 \right)</math> is greater than <math>\frac12</math> can be written as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||
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+ | ==Solution== | ||
+ | The areas bounded by the unit square and alternately bounded by the lines through <math>\left(\frac{5}{8},\frac{3}{8}\right)</math> that are vertical or have a slope of <math>1/2</math> show where <math>P</math> can be placed to satisfy the condition. One of the areas is a trapezoid with bases <math>1/16</math> and <math>3/8</math> and height <math>5/8</math>. The other area is a trapezoid with bases <math>7/16</math> and <math>5/8</math> and height <math>3/8</math>. Then, <cmath>\frac{\frac{1}{16}+\frac{3}{8}}{2}\cdot\frac{5}{8}+\frac{\frac{7}{16}+\frac{5}{8}}{2}\cdot\frac{3}{8}=\frac{86}{256}=\frac{43}{128}\implies43+128=\boxed{171}</cmath> | ||
+ | ~mn28407 | ||
==Video Solution== | ==Video Solution== |
Revision as of 02:05, 8 June 2020
Contents
[hide]Problem
Let be a point chosen uniformly at random in the interior of the unit square with vertices at , and . The probability that the slope of the line determined by and the point is greater than can be written as , where and are relatively prime positive integers. Find .
Solution
The areas bounded by the unit square and alternately bounded by the lines through that are vertical or have a slope of show where can be placed to satisfy the condition. One of the areas is a trapezoid with bases and and height . The other area is a trapezoid with bases and and height . Then, ~mn28407
Video Solution
https://youtu.be/x0QznvXcwHY?t=190
~IceMatrix
See Also
2020 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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