Difference between revisions of "1996 AJHSME Problems/Problem 11"
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* [[AJHSME Problems and Solutions]] | * [[AJHSME Problems and Solutions]] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
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Latest revision as of 23:24, 4 July 2013
Problem
Let 
be the number
![\[0.\underbrace{0000...0000}_{1996\text{ zeros}}1,\]](http://latex.artofproblemsolving.com/0/6/d/06db9469a513bb08b2ce3a07853519ebac9fc909.png)
where there are 1996 zeros after the decimal point. Which of the following expressions represents the largest number?
Solution
Estimate each of the options.
will give a number that is just over 
.
will give a number that is just under . This eliminates , because is bigger.
will give a number that is barely over 
, since it is three times a tiny number. This eliminates , because is bigger.
will give a huge number. 
will get very, very large in magnitude when gets close to zero. You can see this by examining the sequence 
, which gives 
as the reciprocal, 
, which gives 
as the reciprocal, and 
, which gives 
as the reciprocal. Thus, will be huge, and this eliminates .
will give a small number, since you're dividing a tiny number into thirds. This eliminates , and gives 
as the answer.
See Also
| 1996 AJHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 10 |
Followed by Problem 12 | |
| 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 AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
