2021 Fall AMC 10B Problems/Problem 19
Contents
Problem
Let be the positive integer , a -digit number where each digit is a . Let be the leading digit of the th root of . What is
Solution 1
We can rewrite as . When approximating values, as we will shortly do, the minus one will become negligible so we can ignore it. When we take the power of ten out of the square root, we’ll be multiplying by another power of ten, so the leading digit will not change. Thus the leading digit of will be equal to the leading digit of .
Then is the first digit of
.
.
.
.
The final answer is therefore .
~KingRavi
Solution 2
For notation purposes, let be the number with digits, and let be the leading digit of . As an example, , because , and the first digit of that is .
Notice that for all numbers ; this is because , and dividing by does not affect the leading digit of a number. Similarly, In general, for positive integers and real numbers , it is true that Behind all this complex notation, all that we're really saying is that the first digit of something like has the same first digit as and .
The problem asks for
From our previous observation, we know that Therefore, . We can evaluate , the leading digit of , to be . Therefore, .
Similarly, we have Therefore, . We know , so .
Next, and , so .
We also have and , so .
Finally, and , so .
We have that .
~ihatemath123
Solution 3 (Condensed Solution 1)
Since is a digit number and is around , we have is . is the same story, so is . It is the same as as well, so is also . However, is mod , so we need to take the 5th root of , which is between and , and therefore, is . is the same as , since it is more than a multiple of . Therefore, we have which is .
~Arcticturn
Solution 4
First, we compute .
Because , . Because , .
Therefore, .
Second, we compute .
Because , . Because , .
Therefore, .
Third, we compute .
Because , . Because , .
Therefore, .
Fourth, we compute .
Because , . Because , .
Therefore, .
Fifth, we compute .
Because , . Because , .
Therefore, .
Therefore,
Therefore, the answer is .
~Steven Chen (www.professorchenedu.com)
Solution 5 (Guessing)
Benford's Law states that in random numbers, the leading digit is more likely to be or rather than or . From here, we can eliminate C, D, E. It is better to guess between A and B than not guess at all since your expected score from doing this is points.
~MathFun1000
Video Solution by Interstigation
~Interstigation
Video Solution 2 by WhyMath
~savannahsolver
See Also
2021 Fall AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
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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