Art of Problem Solving

2021 WSMO Speed Round/Problem 1: Difference between revisions

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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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==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*}
|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}
\end{align*}

Latest revision as of 16:47, 22 December 2021