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...")
 
(Solution (bash))
Line 4: Line 4:
 
==Solution (bash)==
 
==Solution (bash)==
 
Note that  
 
Note that  
\begin{align*}
+
<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)|&=f(f(f(f(f(f(f(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>
&=f(f(f(f(f(f((2-1)^2))))))\
+
<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>
&=f(f(f(f(f(f(1))))))\
 
&=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)))\
 
&=f(f(f(0)))\
 
&=f(f((0-1)^2))\
 
&=f(f(1))\
 
&=f((1-1)^2)\
 
&=f(0)\
 
&=(0-1)^2=\boxed{1}
 
\end{align*}
 

Revision as of 16:33, 22 December 2021

Problem

Let $f^1(x)=(x-1)^2$, and let $f^n(x)=f^1(f^{n-1}(x))$. Find the value of $|f^7(2)|$.

Solution (bash)

Note that \[|f^7(2)|=f(f(f(f(f(f(f(2)))))))=f(f(f(f(f(f((2-1)^2))))))=f(f(f(f(f(f(1))))))\] \[=f(f(f(f(f((1-1)^2)))))=f(f(f(f(f(0)))))=f(f(f(f((0-1)^2))))=f(f(f(f(1))))=f(f(f((1-1)^2)))\] \[=f(f(f(0)))=f(f((0-1)^2))=f(f(1))=f((1-1)^2)=f(0)=(0-1)^2=\boxed{1}\]