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Difference between revisions of "1990 AIME Problems/Problem 13"

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Let's divide all elements of T into sections. Each section ranges from <math>10^{i}</math> to <math>10^{i+1}</math> And, each section must have 1 or 2 elements. So, let's consider both cases.  
 
Let's divide all elements of T into sections. Each section ranges from <math>10^{i}</math> to <math>10^{i+1}</math> And, each section must have 1 or 2 elements. So, let's consider both cases.  
  
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If a section has one element, we claim that the number doesn't have 9 as the leftmost digit. Let this element be <math>9^{k}</math> and the section ranges from <math>10^{i}</math> to <math>10^{i+1}</math>. To the contrary, let's assume the number (<math>9^{k}</math>) does have 9 as the leftmost digit. Thus, <math>9 \cdot 10^i \leq 9^k</math>. But, if you divide both sides by 9, you get <math>10^i \leq 9^{k-1}</math>, and because <math>9^{k-1} < 9^{k} \leq 10^{i+1}</math>, so we have another number (<math>9^{k-1}</math>) in the same section (<math>10^i \leq 9^{k-1} < 10^{i+1}</math>). Which is a contradiction to our assumption that the section only has 1 element. So in this case, the number doesn't have 9 as the leftmost digit.
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If a section has one element, we claim that the number doesn't have 9 as the leftmost digit. Let this element be <math>9^{k}</math> and the section ranges from <math>10^{i}</math> to <math>10^{i+1}</math>. To the contrary, let's assume the number (<math>9^{k}</math>) does have 9 as the leftmost digit. Thus, <math>9 \cdot 10^i \leq 9^k</math>. But, if you divide both sides by 9, you get <math>10^i \leq 9^{k-1}</math>, and because <math>9^{k-1} < 9^{k} < 10^{i+1}</math>, so we have another number (<math>9^{k-1}</math>) in the same section (<math>10^i \leq 9^{k-1} < 10^{i+1}</math>). Which is a contradiction to our assumption that the section only has 1 element. So in this case, the number doesn't have 9 as the leftmost digit.
  
 
If a section has 2 elements, we claim one has to have a 9 as the leftmost digit, one doesn't. Let the elements be <math>9^{k}</math> and <math>9^{k+1}</math>, and the section ranges from <math>10^{i}</math> to <math>10^{i+1}</math>. We know that <math>10^{i} \leq 9^{k} </math>, and thus <math>9 \cdot 10^{i} \leq 9^{k+1} </math>, and since <math>9^{k+1} \leq 10^{i+1}</math>, it's leftmost digit must be 9, and the other number's leftmost digit is 1.
 
If a section has 2 elements, we claim one has to have a 9 as the leftmost digit, one doesn't. Let the elements be <math>9^{k}</math> and <math>9^{k+1}</math>, and the section ranges from <math>10^{i}</math> to <math>10^{i+1}</math>. We know that <math>10^{i} \leq 9^{k} </math>, and thus <math>9 \cdot 10^{i} \leq 9^{k+1} </math>, and since <math>9^{k+1} \leq 10^{i+1}</math>, it's leftmost digit must be 9, and the other number's leftmost digit is 1.

Revision as of 14:27, 27 June 2019

Problem

Let $T = \{9^k : k ~ \mbox{is an integer}, 0 \le k \le 4000\}$. Given that $9^{4000}_{}$ has 3817 digits and that its first (leftmost) digit is 9, how many elements of $T_{}^{}$ have 9 as their leftmost digit?

Solution 1

Since $9^{4000}$ has 3816 digits more than $9^1$, $4000 - 3816 = \boxed{184}$ numbers have 9 as their leftmost digits.

Solution 2

Let's divide all elements of T into sections. Each section ranges from $10^{i}$ to $10^{i+1}$ And, each section must have 1 or 2 elements. So, let's consider both cases.

If a section has one element, we claim that the number doesn't have 9 as the leftmost digit. Let this element be $9^{k}$ and the section ranges from $10^{i}$ to $10^{i+1}$. To the contrary, let's assume the number ($9^{k}$) does have 9 as the leftmost digit. Thus, $9 \cdot 10^i \leq 9^k$. But, if you divide both sides by 9, you get $10^i \leq 9^{k-1}$, and because $9^{k-1} < 9^{k} < 10^{i+1}$, so we have another number ($9^{k-1}$) in the same section ($10^i \leq 9^{k-1} < 10^{i+1}$). Which is a contradiction to our assumption that the section only has 1 element. So in this case, the number doesn't have 9 as the leftmost digit.

If a section has 2 elements, we claim one has to have a 9 as the leftmost digit, one doesn't. Let the elements be $9^{k}$ and $9^{k+1}$, and the section ranges from $10^{i}$ to $10^{i+1}$. We know that $10^{i} \leq 9^{k}$, and thus $9 \cdot 10^{i} \leq 9^{k+1}$, and since $9^{k+1} \leq 10^{i+1}$, it's leftmost digit must be 9, and the other number's leftmost digit is 1.

There are total 4001 elements and 3817 sections (each esction can only has one or two elements). The two elements section must have 9 as the left most digit. We can take 4001 elements, subtract 3817 elements that don't have 9 as the leftmost digit, and get $\boxed{184}$ numbers that have 9 as the leftmost digit.

- AlexLikeMath

See also

1990 AIME (Problems • Answer Key • Resources)
Preceded by
Problem 12 		Followed by
Problem 14
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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