2020 AMC 10B Problems/Problem 12
Contents
Problem
The decimal representation of 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
We have
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 as long as n is not a power of 10, 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 (Alternate 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 (Scientific Notation)
We see that . We see that this has zeros after the decimal point before coming to .
Therefore, the answer is
Solution 6 (Logarithms Without Words)
~phoenixfire
==Video Solution (HOW TO CRITICALLY THINK) https://youtu.be/qnznImCDSW4
~Education, the Study of Everything
Solution 7 (Algebra)
Define as the number of consecutive s after the decimal point in the decimal representation of Thus, has zeroes followed by a when expressed as a decimal, and is greater than So, we have that Taking the reciprocal and switching signs grants We can express as We know that Multiplying this with adds more zeroes at the end, which means the left side of our inequality has digits. Then, the on the right side must have an exponent less than meaning that
We are also given that the initial string of zeroes in the decimal representation of is followed by a If we consider just these digits after the decimal point, we can express it as the fraction Since there are more digits after this in the decimal representation of it is true that Rinse and repeat and we have Multiplying both sides by gives We previously found that which has digits. Multiplying this by will keep the number of digits the same (make sure to see why). Thus,
If and then
~happyhari, mathbrek
Video Solution
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 |
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