Difference between revisions of "2020 AMC 12B Problems/Problem 25"
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<cmath>\sin^{2}{(\pi x)}+\sin^{2}{(\pi y)}>1</cmath> | <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> | + | <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> | + | 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> |
| + | |||
| + | <cmath>\sin{(\pi x)}=\cos{(\pi y)}=sin{(\frac{\pi}{2}\pm \pi y)}</cmath> | ||
Revision as of 23:36, 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/a/3/e/a3e20e036a736e493cf3d41f17f106a79f6ebc86.png)
Let's consider the boundary cases: 
and 
![\[\sin{(\pi x)}=\cos{(\pi y)}=sin{(\frac{\pi}{2}\pm \pi y)}\]](http://latex.artofproblemsolving.com/6/e/4/6e43462afbdf801f7d203dafa795c14338838ea0.png)
