Difference between revisions of "2020 AIME II Problems/Problem 2"
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| − | + | ==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 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== | ||
| + | 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 | ||
| + | |||
| + | ==Solution 2 (Official MAA)== | ||
| + | The line through the fixed point <math>\left(\frac58,\frac38\right)</math> with slope <math>\frac12</math> has equation <math>y=\frac12 x + \frac1{16}</math>. The slope between <math>P</math> and the fixed point exceeds <math>\frac12</math> if <math>P</math> falls within the shaded region in the diagram below consisting of two trapezoids with area | ||
| + | <cmath>\frac{\frac1{16}+\frac38}2\cdot\frac58 + \frac{\frac58+\frac7{16}}2\cdot\frac38 = \frac{43}{128}.</cmath>Because the entire square has area <math>1,</math> the required probability is <math>\frac{43}{128}</math>. The requested sum is <math>43+128 = 171</math>. | ||
| + | <asy> | ||
| + | defaultpen(fontsize(8pt)); | ||
| + | unitsize(4cm); | ||
| + | pair A = (0,0); | ||
| + | pair B = (1,0); | ||
| + | pair C = (1,1); | ||
| + | pair D = (0,1); | ||
| + | |||
| + | pair F = (0, 1/16); | ||
| + | pair G = (1, 9/16); | ||
| + | pair H = (5/8, 0); | ||
| + | pair J = (5/8, 1); | ||
| + | pair K = IP(H--J, F--G); | ||
| + | |||
| + | pair P = (13/16, 12/16); | ||
| + | pair Q = extension(P,K,A,B); | ||
| + | pair R = extension(K,P,C,D); | ||
| + | |||
| + | draw(A--B--C--D--cycle); | ||
| + | |||
| + | label("$(0,0)$", A, SW); | ||
| + | label("$(1,0)$", B, SE); | ||
| + | label("$(1,1)$", C, E); | ||
| + | label("$(0,1)$", D, W); | ||
| + | |||
| + | filldraw(A--H--K--F--cycle, lightgray); | ||
| + | filldraw(K--G--C--J--cycle, lightgray); | ||
| + | |||
| + | dot(K); | ||
| + | dot("$P$", P, W); | ||
| + | draw(Q -- R, dashed); | ||
| + | |||
| + | label("$\frac 38$", H--K, E); | ||
| + | label("$\frac 58$", K--J, W); | ||
| + | label("$\frac 7{16}$", G--C, E); | ||
| + | label("$\frac 38$", C--J, N); | ||
| + | label("$\frac 1{16}$", A--F, dir(160)); | ||
| + | |||
| + | </asy> | ||
| + | |||
| + | ==Video Solution== | ||
| + | https://youtu.be/x0QznvXcwHY?t=190 | ||
| + | |||
| + | ~IceMatrix | ||
| + | |||
| + | |||
| + | ==Video Solution 2== | ||
| + | https://youtu.be/VBwlVbM0GRw | ||
| + | |||
| + | ~avn | ||
| + | |||
| + | ==See Also== | ||
| + | {{AIME box|year=2020|n=II|num-b=1|num-a=3}} | ||
| + | [[Category:Intermediate Geometry Problems]] | ||
| + | {{MAA Notice}} | ||
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, ![\[\frac{\frac{1}{16}+\frac{3}{8}}{2}\cdot\frac{5}{8}+\frac{\frac{7}{16}+\frac{5}{8}}{2}\cdot\frac{3}{8}=\frac{86}{256}=\frac{43}{128}\implies43+128=\boxed{171}\]](http://latex.artofproblemsolving.com/2/a/4/2a4462215b3f5d9537b3bf6cf1c7aed20b86b183.png)
~mn28407
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
![\[\frac{\frac1{16}+\frac38}2\cdot\frac58 + \frac{\frac58+\frac7{16}}2\cdot\frac38 = \frac{43}{128}.\]](http://latex.artofproblemsolving.com/1/d/a/1dacf65fa2ac2d45dea84143eb934e3ee347cdd3.png)
Because the entire square has area 
the required probability is 
. The requested sum is 
.
![[asy] defaultpen(fontsize(8pt)); unitsize(4cm); pair A = (0,0); pair B = (1,0); pair C = (1,1); pair D = (0,1); pair F = (0, 1/16); pair G = (1, 9/16); pair H = (5/8, 0); pair J = (5/8, 1); pair K = IP(H--J, F--G); pair P = (13/16, 12/16); pair Q = extension(P,K,A,B); pair R = extension(K,P,C,D); draw(A--B--C--D--cycle); label("$(0,0)$", A, SW); label("$(1,0)$", B, SE); label("$(1,1)$", C, E); label("$(0,1)$", D, W); filldraw(A--H--K--F--cycle, lightgray); filldraw(K--G--C--J--cycle, lightgray); dot(K); dot("$P$", P, W); draw(Q -- R, dashed); label("$\frac 38$", H--K, E); label("$\frac 58$", K--J, W); label("$\frac 7{16}$", G--C, E); label("$\frac 38$", C--J, N); label("$\frac 1{16}$", A--F, dir(160)); [/asy]](http://latex.artofproblemsolving.com/9/6/5/965b0d52cd7d196c2bcdbfbe72c01a25560ae5bc.png)
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
