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Difference between revisions of "2011 AMC 12B Problems/Problem 17"

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==Solution==
 
==Solution==
  
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<math>g(x)\text{ = }\text{log}_{10}\left(\frac{x}{10}\right)\text{ = }\text{log}_{10}\left({x}\right)\text{ - 1}</math>
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<math>g(x)=\log_{10}\left(\frac{x}{10}\right)=\log_{10}\left({x}\right) - 1</math>
  
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<math>h_{1}(x)\text{ = }g(f(x))\text{ = }g(10^{10x})\text{ = }\text{log}_{10}\left({10^{10x}}\right){ - 1 = 10x - 1}</math>
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<math>h_{1}(x)=g(f(x))\text{ = }g(10^{10x})=\log_{10}\left({10^{10x}}\right){ - 1 = 10x - 1}</math>
  
 
Proof by induction that <math>h_{n}(x)\text{ = }10^n x - (1 + 10 + 10^2 + ... + 10^{n-1})</math>:
 
Proof by induction that <math>h_{n}(x)\text{ = }10^n x - (1 + 10 + 10^2 + ... + 10^{n-1})</math>:
  
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For <math>\text{n = 1, }h_{1}(x)\text{ = }10x - 1</math>
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For <math>n=1</math>, <math>h_{1}(x)=10x - 1</math>
  
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Assume <math>h_{n}(x)\text{ = }10^n x - (1 + 10 + 10^2 + ... + 10^{n-1})</math> is true for n:
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Assume <math>h_{n}(x)=10^n x - (1 + 10 + 10^2 + ... + 10^{n-1})</math> is true for n:
  
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<math>h_{n+1}(x)\text{ = } h_{1}(h_{n}(x))\text{ = }10 h_{n}(x) - 1\text{ = 10 }(10^n x  - (1 + 10 + 10^2 + ... + 10^{n-1})) - 1
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<cmath>\begin{align*}
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\\= 10^{n+1} x - (10 + 10^2 + ... + 10^{n}) - 1
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h_{n+1}(x)&= h_{1}(h_{n}(x))\\
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\\= 10^{n+1} x - (1 + 10 + 10^2 + ... + 10^{(n+1)-1})</math>
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&=10 h_{n}(x) - 1\\
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&=10 (10^n x  - (1 + 10 + 10^2 + ... + 10^{n-1})) - 1\\
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&= 10^{n+1} x - (10 + 10^2 + ... + 10^{n}) - 1\\
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&= 10^{n+1} x - (1 + 10 + 10^2 + ... + 10^{(n+1)-1})
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\end{align*}</cmath>
  
 
Therefore, if it is true for n, then it is true for n+1; since it is also true for n = 1, it is true for all positive integers n.
 
Therefore, if it is true for n, then it is true for n+1; since it is also true for n = 1, it is true for all positive integers n.
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The sum of the digits is 8 times 2010 plus 9, or <math>\boxed{16089\textbf{(B)}}</math>
 
The sum of the digits is 8 times 2010 plus 9, or <math>\boxed{16089\textbf{(B)}}</math>
 +
 +
==Solution 2 (Quick, Non-Rigorous Trends)==
 +
As before, <math>h_1(x)=10x-1</math>. Compute <math>h_1(x)</math>, <math>h_2(x)</math>, and <math>h_3(x)</math> to yield 9, 89, and 889. Notice how this trend will repeat this trend (multiply by 10, subtract 1, repeat). As such, <math>h_{2011}</math> is just 2010 8's followed by a nine. <math>2010(8)+9=\boxed{\textbf{B)}16089}</math>.
 +
 +
~~BJHHar
  
 
== See also ==
 
== See also ==
 
{{AMC12 box|year=2011|num-b=16|num-a=18|ab=B}}
 
{{AMC12 box|year=2011|num-b=16|num-a=18|ab=B}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 15:40, 21 September 2024

Problem

Let $f(x) = 10^{10x}, g(x) = \log_{10}\left(\frac{x}{10}\right), h_1(x) = g(f(x))$, and $h_n(x) = h_1(h_{n-1}(x))$ for integers $n \geq 2$. What is the sum of the digits of $h_{2011}(1)$?

$\textbf{(A)}\ 16081 \qquad \textbf{(B)}\ 16089 \qquad \textbf{(C)}\ 18089 \qquad \textbf{(D)}\ 18098 \qquad \textbf{(E)}\ 18099$

Solution

$g(x)=\log_{10}\left(\frac{x}{10}\right)=\log_{10}\left({x}\right) - 1$

$h_{1}(x)=g(f(x))\text{ = }g(10^{10x})=\log_{10}\left({10^{10x}}\right){ - 1 = 10x - 1}$

Proof by induction that $h_{n}(x)\text{ = }10^n x - (1 + 10 + 10^2 + ... + 10^{n-1})$:

For $n=1$, $h_{1}(x)=10x - 1$

Assume $h_{n}(x)=10^n x - (1 + 10 + 10^2 + ... + 10^{n-1})$ is true for n:

\begin{align*} h_{n+1}(x)&= h_{1}(h_{n}(x))\\ &=10 h_{n}(x) - 1\\ &=10 (10^n x - (1 + 10 + 10^2 + ... + 10^{n-1})) - 1\\ &= 10^{n+1} x - (10 + 10^2 + ... + 10^{n}) - 1\\ &= 10^{n+1} x - (1 + 10 + 10^2 + ... + 10^{(n+1)-1}) \end{align*}

Therefore, if it is true for n, then it is true for n+1; since it is also true for n = 1, it is true for all positive integers n.

$h_{2011}(1) = 10^{2011}\times 1{ - }(1 + 10 + 10^2 + ... + 10^{2010})$, which is the 2011-digit number 8888...8889

The sum of the digits is 8 times 2010 plus 9, or $\boxed{16089\textbf{(B)}}$

Solution 2 (Quick, Non-Rigorous Trends)

As before, $h_1(x)=10x-1$. Compute $h_1(x)$, $h_2(x)$, and $h_3(x)$ to yield 9, 89, and 889. Notice how this trend will repeat this trend (multiply by 10, subtract 1, repeat). As such, $h_{2011}$ is just 2010 8's followed by a nine. $2010(8)+9=\boxed{\textbf{B)}16089}$.

~~BJHHar

See also

2011 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
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 12 Problems and Solutions

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