Difference between revisions of "2020 AMC 12B Problems/Problem 25"
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==Solution== | ==Solution== | ||
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| + | Let's start first by manipulating the given inequality. | ||
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| + | <cmath>\sin^{2}{(\pi x)}+\sin^{2}{(\pi y)}>1</cmath> | ||
| + | <cmath>\sin^{2}{\pi x}>1-\sin^{2}{\pi y}=\cos^{2}{\pi y}</cmath> | ||
| + | |||
| + | Let's consider the boundary cases: <math>\sin^{2}{\pi x}=\cos^{2}{\pi y}</math> and <math>\sin^{2}{\pi x}=-\cos^{2}{\pi y}</math> | ||
Revision as of 23:34, 7 February 2020
Problem 25
For each real number 
with 
, let numbers 
and 
be chosen independently at random from the intervals ![$[0, a]$](http://latex.artofproblemsolving.com/c/e/7/ce7d05d4da0d50abf77ee166ae0c6474bee5a1cf.png)
and ![$[0, 1]$](http://latex.artofproblemsolving.com/a/b/1/ab178d831a786b92cb4c9ddc2d33578223036f98.png)
, respectively, and let 
be the probability that
![\[\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1\]](http://latex.artofproblemsolving.com/4/d/d/4dda018877a7b217408e61c488287f3bb9d31582.png)
What is the maximum value of 
Solution
Let's start first by manipulating the given inequality.
![\[\sin^{2}{(\pi x)}+\sin^{2}{(\pi y)}>1\]](http://latex.artofproblemsolving.com/c/b/8/cb8efe9949fa008c546e52b562015960b29b76f7.png)
![\[\sin^{2}{\pi x}>1-\sin^{2}{\pi y}=\cos^{2}{\pi y}\]](http://latex.artofproblemsolving.com/4/2/b/42bc7ac66fa0d0d9ce3e750b2ec2d0610a95926e.png)
Let's consider the boundary cases: 
and 
