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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>
+
<math>g(x)=\log_{10}\left(\frac{x}{10}\right)=\log_{10}\left({x}\right) - 1</math>
  
−
<math>h_{1}(x)\text{ = }g(f(x))\text{ = }g(10^{10x})\text{ = }\text{log}_{10}\left({10^{10x}}\right){ - 1 = 10x - 1}</math>
+
<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>:
  
−
For <math>\text{n = 1, }h_{1}(x)\text{ = }10x - 1</math>
+
For <math>n=1</math>, h_{1}(x)=10x - 1<math>
  
−
Assume <math>h_{n}(x)\text{ = }10^n x - (1 + 10 + 10^2 + ... + 10^{n-1})</math> is true for n:
+
Assume </math>h_{n}(x)=10^n x - (1 + 10 + 10^2 + ... + 10^{n-1})<math> is true for n:
  
−
<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
+
\begin{align*}
−
\\= 10^{n+1} x - (10 + 10^2 + ... + 10^{n}) - 1
+
h_{n+1}(x)&= h_{1}(h_{n}(x))\\
−
\\= 10^{n+1} x - (1 + 10 + 10^2 + ... + 10^{(n+1)-1})</math>
+
&=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.
 
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.
  
−
<math>h_{2011}(1) = 10^{2011}\times 1{ - }(1 + 10 + 10^2 + ... + 10^{2010})</math>, which is the 2011-digit number 8888...8889
+
</math>h_{2011}(1) = 10^{2011}\times 1{ - }(1 + 10 + 10^2 + ... + 10^{2010})<math>, which is the 2011-digit number 8888...8889
  
−
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)}}$
  
 
== 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}}

Revision as of 11:29, 11 October 2018

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.$ (Error compiling LaTeX. Unknown error_msg)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$ (Error compiling LaTeX. Unknown error_msg)\boxed{16089\textbf{(B)}}$

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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