Difference between revisions of "2020 AMC 10B Problems/Problem 12"
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==Solution 5 (Logarithms)== | ==Solution 5 (Logarithms)== | ||
| − | <cmath>\lceil log \dfrac{1}{20^{20}} \rceil | + | <cmath>|\lceil log \dfrac{1}{20^{20}} \rceil| |
| − | = \lceil log 20^{-20} \rceil | + | = |\lceil log 20^{-20} \rceil| |
| − | = \lceil -20 log(20) \rceil | + | = |\lceil -20 log(20) \rceil| |
| − | = \lceil -20(log 10 + log 2) \rceil | + | = |\lceil -20(log 10 + log 2) \rceil| |
| − | = \lceil -20(1 + 0.301) \rceil | + | = |\lceil -20(1 + 0.301) \rceil| |
| − | = \lceil -26.02 \rceil | + | = |\lceil -26.02 \rceil| |
| − | = -26 | + | = |-26| |
| − | + | = \boxed{\textbf{(D) } \text{26}}</cmath> | |
~phoenixfire | ~phoenixfire | ||
Revision as of 13:48, 20 February 2020
Contents
Problem
The decimal representation of![\[\dfrac{1}{20^{20}}\]](http://latex.artofproblemsolving.com/6/c/e/6ce6559c22b38f698ff7b67ddc88f2d4d89c4084.png)
consists of a string of zeros after the decimal point, followed by a 
and then several more digits. How many zeros are in that initial string of zeros after the decimal point?
Solution 1
![\[\dfrac{1}{20^{20}} = \dfrac{1}{(10\cdot2)^{20}}=\dfrac{1}{10^{20}\cdot2^{20}}\]](http://latex.artofproblemsolving.com/a/d/8/ad8d5ac8a5d2aa9c9319a5d7a0f4bb219e791abe.png)
Now we do some estimation. Notice that 
, which means that 
is a little more than 
. Multiplying it with 
, we get that the denominator is about 
. Notice that when we divide 
by an 
digit number, there are 
zeros before the first nonzero digit. This means that when we divide by the 
digit integer , there are 
zeros in the initial string after the decimal point. -PCChess
Solution 2
First rewrite 
as 
. Then, we know that when we write this in decimal form, there will be 40 digits after the decimal point. Therefore, we just have to find the number of digits in 
.
and memming 
(alternatively use the fact that 
), 
digits.
Our answer is .
Solution 3 (Brute Force)
Just as in Solution 
we rewrite 
as 
We then calculate 
entirely by hand, first doing 
then multiplying that product by itself, resulting in 
Because this is 
digits, after dividing this number by 
fourteen times, the decimal point is before the 
Dividing the number again by twenty-six more times allows a string of zeroes to be formed. -OreoChocolate
Solution 4 (Smarter Brute Force)
Just as in Solutions 
and 
we rewrite as We can then look at the number of digits in powers of 
. 
, 
, 
, 
, 
, 
, 
and so on. We notice after a few iterations that every power of five with an exponent of 
, the number of digits doesn't increase. This means should have 
digits since there are 
numbers which are from 
to 
, or digits total. This means our expression can be written as 
, where 
is in the range 
. Canceling gives 
, or 
zeroes before the since the number should start on where the one would be in 
. ~aop2014
Solution 5 (Logarithms)
![\[|\lceil log \dfrac{1}{20^{20}} \rceil| = |\lceil log 20^{-20} \rceil| = |\lceil -20 log(20) \rceil| = |\lceil -20(log 10 + log 2) \rceil| = |\lceil -20(1 + 0.301) \rceil| = |\lceil -26.02 \rceil| = |-26| = \boxed{\textbf{(D) } \text{26}}\]](http://latex.artofproblemsolving.com/9/f/5/9f5e651db39f886fb2df98d06734b8d0677c1023.png)
~phoenixfire
Video Solution
~IceMatrix
See Also
| 2020 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 11 |
Followed by Problem 13 | |
| 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 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
