Difference between revisions of "2004 AIME I Problems/Problem 15"
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== Problem == | == Problem == | ||
| − | For all positive integers <math> x | + | For all positive integers <math>x</math>, let |
| − | + | <cmath> | |
| − | < | + | 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} |
| − | + | </cmath> | |
| − | + | 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 | ||
== Solution == | == Solution == | ||
| Line 11: | Line 10: | ||
== See also == | == See also == | ||
| − | + | {{AIME box|year=2004|n=I|num-b=14|after=Last Question}} | |
| − | |||
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Revision as of 15:24, 27 April 2008
Problem
For all positive integers 
, 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: 
and 
for all positive integers 
. Let 
be the smallest such that 
. (For example, 
and 
.) Let 
be the number of positive integers such that 
. Find the sum of the distinct prime factors of .
Solution
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See also
| 2004 AIME I (Problems • Answer Key • Resources) | ||
| 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 | ||
