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 or equal to <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>. |
==Solution== | ==Solution== |
Latest revision as of 16:24, 24 June 2020
Contents
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 or equal to 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
Solution 2 (Official MAA)
The line through the fixed point with slope has equation . The slope between and the fixed point exceeds if falls within the shaded region in the diagram below consisting of two trapezoids with area Because the entire square has area the required probability is . The requested sum is .
Video Solution
https://youtu.be/x0QznvXcwHY?t=190
~IceMatrix
Video Solution 2
~avn
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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