2013 AMC 12A Problems/Problem 21
Contents
Problem
Consider . Which of the following intervals contains ?
Solutions
Solution 1
Let and , and from the problem description,
We can reason out an approximation, by ignoring the :
And a better approximation, by plugging in our first approximation for in our original definition for :
And an even better approximation:
Continuing this pattern, obviously, will eventually terminate at , in other words our original definition of .
However, at , going further than will not distinguish between our answer choices. is nearly indistinguishable from .
So we take and plug in.
Since , we know . This gives us our answer range:
Solution 2
Suppose . Then . So if , then . So . Repeating, we then get . This is clearly absurd (the RHS continues to grow more than exponentially for each iteration). So, is not greater than . So . But this leaves only one answer, so we are done.
Solution 3
Define , and We are looking for . First we show
Lemma. For any integer , if then .
Proof. First note that . Let . Then , so . Suppose the claim is true for . Then . The Lemma is thus proved by induction.
Finally, note that so that the Lemma implies that . This means that , which leaves us with only one option .
Solution 4
Define , and We start with a simple observation:
Lemma. For , .
Proof. Since , we have , so .
It follows that , and so on.
Thus .
Then .
It follows that .
Finally, we get , which leaves us with only option .
Solution 5 (nonrigorous + abusing answer choices.)
Intuitively, you can notice that , therefore (by the answer choices) . We can then say:
The only answer choice that is possible given this information is
Solution 6 (super quick)
Let . From the answer choices, we see that . Since grows very slowly, we can assume . Therefore, .
~rayfish
Solution 7 (Not rigorous but it works)
Start small from . Because is close to , the expression is close to . This continues for the 1 digit numbers. However, when we get to , so the expression equals . We can round down how much each new log contributes to the expression as an approximate answer. There are 90 2 digit numbers, 900 3 digit numbers, and 1014 4 digit numbers that increase the expression. , so it increases between 3 and 4. The answer is .
~dragnin
Solution 8 (Assumption)
Consider and disregard every other log. We do this because we assume that the logs from onward cannot contribute more than because of the nature of logs. We have so the answer is
Video Solution by Richard Rusczyk
https://artofproblemsolving.com/videos/amc/2013amc12a/360
See Also
2013 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 20 |
Followed by Problem 22 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.