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MATH / For each x in [0,1], define

Problem

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problem

For each xx in [0,1][0,1], define \[\begin{cases} f(x) = 2x, \qquad\qquad \mathrm{if} \quad 0 \leq x \leq \frac{1}{2};\\ f(x) = 2-2x, \qquad \mathrm{if} \quad \frac{1}{2} < x \leq 1. \end{cases}\]Let f[2](x)=f(f(x))f^{[2]}(x) = f(f(x)), and f[n+1](x)=f[n](f(x))f^{[n + 1]}(x) = f^{[n]}(f(x)) for each integer n≥2n \geq 2. Then the number of values of xx in [0,1][0,1] for which f[2005](x)=12f^{[2005]}(x) = \frac {1}{2} can be expressed in the form pa,p^a, where pp is a prime and aa is a positive integer. Find p+a.p + a.
Plain-text mathematical notation (without MathML)
For each x in [0,1], define
\[\begin{cases}  f(x) = 2x, \qquad\qquad \mathrm{if} \quad 0 \leq x \leq \frac{1}{2};\\  f(x) = 2-2x, \qquad \mathrm{if} \quad \frac{1}{2} < x \leq 1.  \end{cases}\]Let f^([2])(x)=f(f(x)), and f^([n+1])(x)=f^([n])(f(x)) for each integer n≥2.  Then the number of values of x in [0,1] for which f^([2005])(x)=(1)/(2) can be expressed in the form p^(a), where p is a prime and a is a positive integer.  Find p+a.
Original LaTeX notation
For each $x$ in $[0,1]$, define
\[\begin{cases}  f(x) = 2x, \qquad\qquad \mathrm{if} \quad 0 \leq x \leq \frac{1}{2};\\  f(x) = 2-2x, \qquad \mathrm{if} \quad \frac{1}{2} < x \leq 1.  \end{cases}\]Let $f^{[2]}(x) = f(f(x))$, and $f^{[n + 1]}(x) = f^{[n]}(f(x))$ for each integer $n \geq 2$.  Then the number of values of $x$ in $[0,1]$ for which $f^{[2005]}(x) = \frac {1}{2}$ can be expressed in the form $p^a,$ where $p$ is a prime and $a$ is a positive integer.  Find $p + a.$

level

Level 5

type

Intermediate Algebra

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Source and history

task_id
293abe16-5b49-5749-90dd-7cf5ccbb2caf
task_key
intermediate~5falgebra--test--293abe16-5b49-5749-90dd-7cf5ccbb2caf
task_revision_id
1
config / split
intermediate_algebra / test

Official source

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