Difference between revisions of "2004 AIME I Problems/Problem 15"

m (LaTeX style)
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== Problem ==
 
== Problem ==
For all positive integers <math> x, </math> let
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For all positive integers <math>x</math>, let
 
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<cmath>
<center><math> f(x)=\begin{cases}1 & \rm{if \ x=1}\ \frac x{10} & \rm{ if \ x \ is \ divisible \ by \ 10}\ x+1 & \rm{otherwise}\end{cases} </math> </center>
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f(x)=\begin{cases}1 & \text{if x = 1}}\ \frac x{10} & \text{if x is divisible by 10}\ x+1 & \text{otherwise}\end{cases}
 
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</cmath>
 
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and define a sequence as follows: <math>x_1=x</math> and <math>x_{n+1}=f(x_n)</math> for all positive integers <math>n</math>. Let <math>d(x)</math> be the smallest <math>n</math> such that <math>x_n=1</math>. (For example, <math>d(100)=3</math> and <math>d(87)=7</math>.) Let <math>m</math> be the number of positive integers <math>x</math> such that <math>d(x)=20</math>. Find the sum of the distinct prime factors of <math>m</math>.
and define a sequence as follows: <math> x_1 = x </math> and <math> x_{n+1} = f(x_n) </math> for all positive integers <math> n. </math> Let <math> d(x) </math> be the smallest <math> n </math> such that <math> x_n = 1. </math> (For example, <math> d(100)=3 </math> and <math> d(87)=7. </math>) Let <math> m </math> be the number of positive integers <math> x </math> such that <math> d(x)=20. </math> Find the sum of the distinct prime factors of <math> m. </math>
 
  
 
== Solution ==
 
== Solution ==
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== See also ==
 
== See also ==
* [[2004 AIME I Problems/Problem 14| Previous problem]]
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{{AIME box|year=2004|n=I|num-b=14|after=Last Question}}
 
 
* [[2004 AIME I Problems]]
 

Revision as of 15:24, 27 April 2008

Problem

For all positive integers $x$, let

\[f(x)=\begin{cases}1 & \text{if x = 1}}\\ \frac x{10} & \text{if x is divisible by 10}\\ x+1 & \text{otherwise}\end{cases}\] (Error compiling LaTeX. Unknown error_msg)

and define a sequence as follows: $x_1=x$ and $x_{n+1}=f(x_n)$ for all positive integers $n$. Let $d(x)$ be the smallest $n$ such that $x_n=1$. (For example, $d(100)=3$ and $d(87)=7$.) Let $m$ be the number of positive integers $x$ such that $d(x)=20$. Find the sum of the distinct prime factors of $m$.

Solution

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See also

2004 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 14
Followed by
Last Question
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions