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2021 WSMO Speed Round/Problem 1

Revision as of 16:28, 22 December 2021 by Pinkpig (talk | contribs) (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 $f^1(x)=(x-1)^2$, and let $f^n(x)=f^1(f^{n-1}(x))$. Find the value of $|f^7(2)|$.

Solution (bash)

Note that \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*}

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