Difference between revisions of "2020 AMC 12B Problems/Problem 25"
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| − | = | + | ==Problem 25== |
| − | + | For each real number <math>a</math> with <math>0 \leq a \leq 1</math>, let numbers <math>x</math> and <math>y</math> be chosen independently at random from the intervals <math>[0, a]</math> and <math>[0, 1]</math>, respectively, and let <math>P(a)</math> be the probability that | |
| − | <cmath></cmath> | + | <cmath>\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1</cmath> |
| + | What is the maximum value of <math>P(a)?</math> | ||
| + | |||
| + | <math>\textbf{(A)}\ \frac{7}{12} \qquad\textbf{(B)}\ 2 - \sqrt{2} \qquad\textbf{(C)}\ \frac{1+\sqrt{2}}{4} \qquad\textbf{(D)}\ \frac{\sqrt{5}-1}{2} \qquad\textbf{(E)}\ \frac{5}{8}</math> | ||
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| + | [[2020 AMC 12B Problems/Problem 25|Solution]] | ||
Revision as of 00:29, 8 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 
