2020 AIME II Problems/Problem 2
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 
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, ![\[\frac{\frac{1}{16}+\frac{3}{8}}{2}\cdot\frac{5}{8}+\frac{\frac{7}{16}+\frac{5}{8}}{2}\cdot\frac{3}{8}=\frac{86}{256}=\frac{43}{128}\implies43+128=\boxed{171}\]](http://latex.artofproblemsolving.com/2/a/4/2a4462215b3f5d9537b3bf6cf1c7aed20b86b183.png)
~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 | |
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| All AIME Problems and Solutions | ||
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