2009 AMC 12A Problems/Problem 24
Problem
The tower function of twos is defined recursively as follows: and
for
. Let
and
. What is the largest integer
such that
![\[\underbrace{\log_2\log_2\log_2\ldots\log_2B}_{k\text{ times}}\]](http://latex.artofproblemsolving.com/3/0/c/30ce227e6c3bfc6b946422c508d8e508f9b940bf.png)
is defined?
Solution
We just look at the last three logarithms for the moment, and use the fact that . We wish to find:
Now we realize that is much smaller than
. So we approximate this, remembering we have rounded down, as:
![\[\log_2(T(2009)) = T(2008)\]](http://latex.artofproblemsolving.com/f/8/4/f84af66b3e57c9626e411ec9aab287e272e3888c.png)
We have used logarithms so far. Applying
more to the left of our expression, we get
. Then we can apply the logarithm
more times, until we get to
. So our answer is approximately
. But we rounded down, so that means that after
logarithms we get a number slightly greater than
, so we can apply logarithms one more time. We can be sure it is small enough so that the logarithm can only be applied
more time since
is the largest answer choice. So the answer is
.
EDIT: Why use answer choices? @above solution should explain at the end WHY they can be sure.
Alternative Solution
Let where there are
's.
is defined iff
iff
. Note
, so
. Thus, we seek the largest
such that
. Now note that
so satisfies the inequality. Since it is the largest choice, it is the answer.
See also
2009 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 23 |
Followed by Problem 25 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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