Difference between revisions of "2021 WSMO Speed Round/Problem 1"
(Created page with "==Problem== Let <math>f^1(x)=(x-1)^2</math>, and let <math>f^n(x)=f^1(f^{n-1}(x))</math>. Find the value of <math>|f^7(2)|</math>. ==Solution (bash)== Note that \begin{align...") |
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==Solution (bash)== | ==Solution (bash)== | ||
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| − | + | <cmath>|f^7(2)|=f(f(f(f(f(f(f(2)))))))=f(f(f(f(f(f((2-1)^2))))))=f(f(f(f(f(f(1))))))</cmath> | |
| − | |f^7(2)| | + | <cmath>=f(f(f(f(f((1-1)^2)))))=f(f(f(f(f(0)))))=f(f(f(f((0-1)^2))))=f(f(f(f(1))))=f(f(f((1-1)^2)))</cmath> |
| − | + | <cmath>=f(f(f(0)))=f(f((0-1)^2))=f(f(1))=f((1-1)^2)=f(0)=(0-1)^2=\boxed{1}</cmath> | |
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Revision as of 17:33, 22 December 2021
Problem
Let 
, and let 
. Find the value of 
.
Solution (bash)
Note that
![\[|f^7(2)|=f(f(f(f(f(f(f(2)))))))=f(f(f(f(f(f((2-1)^2))))))=f(f(f(f(f(f(1))))))\]](http://latex.artofproblemsolving.com/3/3/d/33d9637d2533ffb876684aa542996fda25518c30.png)
![\[=f(f(f(f(f((1-1)^2)))))=f(f(f(f(f(0)))))=f(f(f(f((0-1)^2))))=f(f(f(f(1))))=f(f(f((1-1)^2)))\]](http://latex.artofproblemsolving.com/1/b/5/1b5fa488afbec394528cda2caed09fec8730b4ed.png)
![\[=f(f(f(0)))=f(f((0-1)^2))=f(f(1))=f((1-1)^2)=f(0)=(0-1)^2=\boxed{1}\]](http://latex.artofproblemsolving.com/3/c/9/3c933505f2e5379e5a8bb0bcd5a2726fb5965c65.png)
