2001 AIME II Problems/Problem 8
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Problem
A certain function 
has the properties that 
for all positive real values of 
, and that 
for 
. Find the smallest for which 
.
Solution
Iterating the condition , we find that 
for positive integers 
. We know the definition of 
from 
, so we would like to express 
. Indeed,
![f(2001) = 729\left[1 - \left| \frac{2001}{729} - 2\right|\right] = 186.](http://alt1.artofproblemsolving.com/Forum/latexrender/pictures/8/c/4/8c4dce885119fe21b893630e9da60a818e066f8c.gif)
We now need the smallest such that 
. The range of 
, is 
. Then 
, and the smallest value of is 
. Then,
![186 = 243\left[1 - \left| \frac{x}{243} - 2\right|\right] \Longrightarrow x = \pm 57 + 2 \cdot 243](http://alt1.artofproblemsolving.com/Forum/latexrender/pictures/5/b/7/5b7de6226df76b4799572e324f7ce6663dc13def.gif)
We want the smaller value of 
.
An alternative approach is to consider the graph of , which repeats every power of 
, and resembles the section from expanded by a factor of .
See also
| 2001 AIME II (Problems • Resources) | ||
| Preceded by Problem 7 | Followed by Problem 9 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
