Difference between revisions of "2012 AMC 12A Problems/Problem 24"

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Rearranging the numbers in the sequence  <math>\{a_k\}_{k=1}^{2011}</math> in decreasing order produces a new sequence  <math>\{b_k\}_{k=1}^{2011}</math>.  What is the sum of all integers <math>k</math>, <math>1\le k \le 2011</math>, such that <math>a_k=b_k?</math>
 
Rearranging the numbers in the sequence  <math>\{a_k\}_{k=1}^{2011}</math> in decreasing order produces a new sequence  <math>\{b_k\}_{k=1}^{2011}</math>.  What is the sum of all integers <math>k</math>, <math>1\le k \le 2011</math>, such that <math>a_k=b_k?</math>
  
<math> \textbf{(A)}\ 671\qquad\textbf{(B)}\ 1006\qquad\textbf{(C)}\ 1341\qquad\textbf{(D)}\ 2011\qquad\textbf{(E)}\2012 </math>
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<math> \textbf{(A)}\ 671\qquad\textbf{(B)}\ 1006\qquad\textbf{(C)}\ 1341\qquad\textbf{(D)}\ 2011\qquad\textbf{(E)}\ 2012 </math>
 
 
  
 
== Solution ==
 
== Solution ==
  
=== Solution 1 ===
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First, we must understand two important functions: <math>f(x) = b^x</math> for <math>0 < b < 1</math>(decreasing exponential function), and <math>g(x) = x^k</math> for <math>k > 0</math>(increasing exponential function for positive <math>x</math>). <math>f(x)</math> is used to establish inequalities when we change the ''exponent'' and keep the ''base'' constant. <math>g(x)</math> is used to establish inequalities when we change the ''base'' and keep the ''exponent'' constant.
We begin our solution by understanding two important functions: <math>f(x) = b^x</math> for <math>0 < b < 1</math>, and <math>g(x) = x^k</math> for <math>k > 0</math>. The first function is a decreasing exponential function. This means that for numbers <math>m > n</math>, <math>f(m) < f(n)</math>. The second function is an increasing function on the interval <math>[0, \infty]</math>. This means that for numbers <math>m > n</math>, <math>g(m) > g(n)</math>. <math>f(x)</math> is used to establish inequalities when we change the exponent keep the base constant. <math>g(x)</math> is used to establish inequalities when we change the base and keep the exponent constant.
 
  
We will now begin by examining the first few terms.
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We will now examine the first few terms.
  
 
Comparing <math>a_1</math> and <math>a_2</math>, <math>0 < a_1 = (0.201)^1 < (0.201)^{a_1} < (0.2011)^{a_1} = a_2 < 1 \Rightarrow 0 < a_1 < a_2 < 1</math>.
 
Comparing <math>a_1</math> and <math>a_2</math>, <math>0 < a_1 = (0.201)^1 < (0.201)^{a_1} < (0.2011)^{a_1} = a_2 < 1 \Rightarrow 0 < a_1 < a_2 < 1</math>.
 +
 +
Therefore, <math>0 < a_1 < a_2 < 1</math>.
  
 
Comparing <math>a_2</math> and <math>a_3</math>, <math>0 < a_3 = (0.20101)^{a_2} < (0.20101)^{a_1} < (0.2011)^{a_1} = a_2 < 1 \Rightarrow 0 < a_3 < a_2 < 1</math>.
 
Comparing <math>a_2</math> and <math>a_3</math>, <math>0 < a_3 = (0.20101)^{a_2} < (0.20101)^{a_1} < (0.2011)^{a_1} = a_2 < 1 \Rightarrow 0 < a_3 < a_2 < 1</math>.
  
Comparing <math>a_1</math> and <math>a_3</math>, <math>0 < a_1 = (0.201)^1 < (0.201)^{a_2} < (0.20101)^{a_2} = a_3 < 1 \Rightarrow 0 < a_1 < a_3 < 0</math>.
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Comparing <math>a_1</math> and <math>a_3</math>, <math>0 < a_1 = (0.201)^1 < (0.201)^{a_2} < (0.20101)^{a_2} = a_3 < 1 \Rightarrow 0 < a_1 < a_3 < 1</math>.
  
 
Therefore, <math>0 < a_1 < a_3 < a_2 < 1</math>.
 
Therefore, <math>0 < a_1 < a_3 < a_2 < 1</math>.
  
Continuing in this manner, it is easy to see a pattern.
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Comparing <math>a_3</math> and <math>a_4</math>, <math>0 < a_3 = (0.20101)^{a_2} < (0.20101)^{a_3} < (0.201011)^{a_3} = a_4 < 1 \Rightarrow 0 < a_3 < a_4 < 1</math>.
 +
 
 +
Comparing <math>a_2</math> and <math>a_4</math>, <math>0 < a_4 = (0.201011)^{a_3} < (0.201011)^{a_1} < (0.2011)^{a_1} = a_2 < 1 \Rightarrow 0 < a_4 < a_2 < 1</math>.
 +
 
 +
Therefore, <math>0 < a_1 < a_3 < a_4 < a_2 < 1</math>.
  
 +
Continuing in this manner, it is easy to see a pattern(see Note 1).
 +
 +
Therefore, the only <math>k</math> when <math>a_k = b_k</math> is when <math>2(k-1006) = 2011 - k</math>. Solving gives <math>\boxed{\textbf{(C)} 1341}</math>.
 +
 +
 +
'''Note 1''':
 
We claim that <math>0 < a_1 < a_3 < ... < a_{2011} < a_{2010} < ... < a_4 < a_2 < 1</math>.
 
We claim that <math>0 < a_1 < a_3 < ... < a_{2011} < a_{2010} < ... < a_4 < a_2 < 1</math>.
  
We will now use induction to prove this statement. (Note that this is not necessary on the AMC):
+
We can use induction to prove this statement. (not necessary for AMC):
  
 +
Base Case: We have already shown the base case above, where <math>0 < a_1 < a_2 < 1</math>.
  
Rearranging in decreasing order gives <math>1 > b_1 = a_2 > b_2 = a_4 > ... > b_{1005} = a_{2010} > b_{1006} = a_{2011} > ... > b_{2010} = a_3 > b_{2011} =  a_1 > 0</math>.
+
Inductive Step:
  
Therefore, the only <math>k</math> when <math>a_k = b_k</math> is when <math>2(k-1006) = 2011 - k</math>. Solving gives <math>\boxed{\textbf{(C)} 1341}</math>.
+
Rearranging in decreasing order gives
 +
<math>1 > b_1 = a_2 > b_2 = a_4 > ... > b_{1005} = a_{2010} > b_{1006} = a_{2011} > ... > b_{2010} = a_3 > b_{2011} =  a_1 > 0</math>.
 +
 
 +
 
 +
==Video Solution by Richard Rusczyk==
 +
https://artofproblemsolving.com/videos/amc/2012amc12a/255
  
 +
~dolphin7
  
 +
==See Also==
 
{{AMC12 box|year=2012|ab=A|num-b=23|num-a=25}}
 
{{AMC12 box|year=2012|ab=A|num-b=23|num-a=25}}
 +
{{MAA Notice}}

Revision as of 13:07, 15 May 2020

Problem

Let $\{a_k\}_{k=1}^{2011}$ be the sequence of real numbers defined by $a_1=0.201,$ $a_2=(0.2011)^{a_1},$ $a_3=(0.20101)^{a_2},$ $a_4=(0.201011)^{a_3}$, and in general,

\[a_k=\begin{cases}(0.\underbrace{20101\cdots 0101}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if }k\text{ is odd,}\\(0.\underbrace{20101\cdots 01011}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if }k\text{ is even.}\end{cases}\]

Rearranging the numbers in the sequence $\{a_k\}_{k=1}^{2011}$ in decreasing order produces a new sequence $\{b_k\}_{k=1}^{2011}$. What is the sum of all integers $k$, $1\le k \le 2011$, such that $a_k=b_k?$

$\textbf{(A)}\ 671\qquad\textbf{(B)}\ 1006\qquad\textbf{(C)}\ 1341\qquad\textbf{(D)}\ 2011\qquad\textbf{(E)}\ 2012$

Solution

First, we must understand two important functions: $f(x) = b^x$ for $0 < b < 1$(decreasing exponential function), and $g(x) = x^k$ for $k > 0$(increasing exponential function for positive $x$). $f(x)$ is used to establish inequalities when we change the exponent and keep the base constant. $g(x)$ is used to establish inequalities when we change the base and keep the exponent constant.

We will now examine the first few terms.

Comparing $a_1$ and $a_2$, $0 < a_1 = (0.201)^1 < (0.201)^{a_1} < (0.2011)^{a_1} = a_2 < 1 \Rightarrow 0 < a_1 < a_2 < 1$.

Therefore, $0 < a_1 < a_2 < 1$.

Comparing $a_2$ and $a_3$, $0 < a_3 = (0.20101)^{a_2} < (0.20101)^{a_1} < (0.2011)^{a_1} = a_2 < 1 \Rightarrow 0 < a_3 < a_2 < 1$.

Comparing $a_1$ and $a_3$, $0 < a_1 = (0.201)^1 < (0.201)^{a_2} < (0.20101)^{a_2} = a_3 < 1 \Rightarrow 0 < a_1 < a_3 < 1$.

Therefore, $0 < a_1 < a_3 < a_2 < 1$.

Comparing $a_3$ and $a_4$, $0 < a_3 = (0.20101)^{a_2} < (0.20101)^{a_3} < (0.201011)^{a_3} = a_4 < 1 \Rightarrow 0 < a_3 < a_4 < 1$.

Comparing $a_2$ and $a_4$, $0 < a_4 = (0.201011)^{a_3} < (0.201011)^{a_1} < (0.2011)^{a_1} = a_2 < 1 \Rightarrow 0 < a_4 < a_2 < 1$.

Therefore, $0 < a_1 < a_3 < a_4 < a_2 < 1$.

Continuing in this manner, it is easy to see a pattern(see Note 1).

Therefore, the only $k$ when $a_k = b_k$ is when $2(k-1006) = 2011 - k$. Solving gives $\boxed{\textbf{(C)} 1341}$.


Note 1: We claim that $0 < a_1 < a_3 < ... < a_{2011} < a_{2010} < ... < a_4 < a_2 < 1$.

We can use induction to prove this statement. (not necessary for AMC):

Base Case: We have already shown the base case above, where $0 < a_1 < a_2 < 1$.

Inductive Step:

Rearranging in decreasing order gives $1 > b_1 = a_2 > b_2 = a_4 > ... > b_{1005} = a_{2010} > b_{1006} = a_{2011} > ... > b_{2010} = a_3 > b_{2011} =  a_1 > 0$.


Video Solution by Richard Rusczyk

https://artofproblemsolving.com/videos/amc/2012amc12a/255

~dolphin7

See Also

2012 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 23
Followed by
Problem 25
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All AMC 12 Problems and Solutions

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