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Difference between revisions of "2022 AIME II Problems/Problem 4"

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There is a positive real number <math>x</math> not equal to either <math>\tfrac{1}{20}</math> or <math>\tfrac{1}{2}</math> such that<cmath>\log_{20x} (22x)=\log_{2x} (202x).</cmath>The value <math>\log_{20x} (22x)</math> can be written as <math>\log_{10} (\tfrac{m}{n})</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
 
There is a positive real number <math>x</math> not equal to either <math>\tfrac{1}{20}</math> or <math>\tfrac{1}{2}</math> such that<cmath>\log_{20x} (22x)=\log_{2x} (202x).</cmath>The value <math>\log_{20x} (22x)</math> can be written as <math>\log_{10} (\tfrac{m}{n})</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
  
−
==Solution==
+
==Solution 1==
 
We could assume a variable <math>v</math> which equals to both <math>\log_{20x} (22x)</math> and <math>\log_{2x} (202x)</math>.
 
We could assume a variable <math>v</math> which equals to both <math>\log_{20x} (22x)</math> and <math>\log_{2x} (202x)</math>.
  
Line 18: Line 18:
  
 
~DSAERF-CALMIT (https://binaryphi.site)
 
~DSAERF-CALMIT (https://binaryphi.site)
 +
 +
==Solution 2==
 +
 +
We have
 +
\begin{align*}
 +
\log_{20x} (22x)
 +
& = \frac{\log_k 22x}{\log_k 20x} \\
 +
& = \frac{\log_k x + \log_k 22}{\log_k x + \log_k 20} .
 +
\end{align*}
 +
 +
We have
 +
\begin{align*}
 +
\log_{2x} (202x)
 +
& = \frac{\log_k 202x}{\log_k 2x} \\
 +
& = \frac{\log_k x + \log_k 202 }{\log_k x + \log_k 2} .
 +
\end{align*}
 +
 +
Because <math>\log_{20x} (22x)=\log_{2x} (202x)</math>, we get
 +
\[
 +
\frac{\log_k x + \log_k 22}{\log_k x + \log_k 20}
 +
= \frac{\log_k x + \log_k 202 }{\log_k x + \log_k 2} .
 +
\]
 +
 +
We denote this common value as <math>\lambda</math>.
 +
 +
By solving the equality <math>\frac{\log_k x + \log_k 22}{\log_k x + \log_k 20} = \lambda</math>, we get <math>\log_k x = \frac{\log_k 22 - \lambda \log_k 20}{\lambda - 1}</math>.
 +
 +
By solving the equality <math>\frac{\log_k x + \log_k 202 }{\log_k x + \log_k 2} = \lambda</math>, we get <math>\log_k x = \frac{\log_k 202 - \lambda \log_k 2}{\lambda - 1}</math>.
 +
 +
By equating these two equations, we get
 +
<cmath>
 +
\[
 +
\frac{\log_k 22 - \lambda \log_k 20}{\lambda - 1}
 +
= \frac{\log_k 202 - \lambda \log_k 2}{\lambda - 1} .
 +
\]
 +
</cmath>
 +
 +
Therefore,
 +
<cmath>
 +
\begin{align*}
 +
\log_{20x} (22x)
 +
& = \lambda \\
 +
& = \frac{\log_k 22 - \log_k 202}{\log_k 20 - \log_k 2} \\
 +
& = \frac{\log_k \frac{11}{101}}{\log_k 10} \\
 +
& = \log_{10} \frac{11}{101} .
 +
\end{align*}
 +
</cmath>
 +
 +
Therefore, the answer is <math>11 + 101 = \boxed{\textbf{(112) }}</math>.
 +
 +
~Steven Chen (www.professorchenedu.com)
  
 
==See Also==
 
==See Also==
 
{{AIME box|year=2022|n=II|num-b=3|num-a=5}}
 
{{AIME box|year=2022|n=II|num-b=3|num-a=5}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 13:31, 18 February 2022

Problem

There is a positive real number $x$ not equal to either $\tfrac{1}{20}$ or $\tfrac{1}{2}$ such that\[\log_{20x} (22x)=\log_{2x} (202x).\]The value $\log_{20x} (22x)$ can be written as $\log_{10} (\tfrac{m}{n})$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution 1

We could assume a variable $v$ which equals to both $\log_{20x} (22x)$ and $\log_{2x} (202x)$.

So that $(20x)^v=22x \textcircled{1}$ and $(2x)^v=202x \textcircled{2}$

Express $\textcircled{1}$ as: $(20x)^v=(2x \cdot 10)^v=(2x)^v \cdot (10^v)=22x \textcircled{3}$

Substitute $\textcircled{2}$ to $\textcircled{3}$: $202x \cdot (10^v)=22x$

Thus, $v=\log_{10} (\frac{22x}{202x})= \log_{10} (\frac{11}{101})$, where $m=11$ and $n=101$.

Therefore, $m+n = \boxed{112}$.

~DSAERF-CALMIT (https://binaryphi.site)

Solution 2

We have \begin{align*} \log_{20x} (22x) & = \frac{\log_k 22x}{\log_k 20x} \\ & = \frac{\log_k x + \log_k 22}{\log_k x + \log_k 20} . \end{align*}

We have \begin{align*} \log_{2x} (202x) & = \frac{\log_k 202x}{\log_k 2x} \\ & = \frac{\log_k x + \log_k 202 }{\log_k x + \log_k 2} . \end{align*}

Because $\log_{20x} (22x)=\log_{2x} (202x)$, we get \[ \frac{\log_k x + \log_k 22}{\log_k x + \log_k 20} = \frac{\log_k x + \log_k 202 }{\log_k x + \log_k 2} . \]

We denote this common value as $\lambda$.

By solving the equality $\frac{\log_k x + \log_k 22}{\log_k x + \log_k 20} = \lambda$, we get $\log_k x = \frac{\log_k 22 - \lambda \log_k 20}{\lambda - 1}$.

By solving the equality $\frac{\log_k x + \log_k 202 }{\log_k x + \log_k 2} = \lambda$, we get $\log_k x = \frac{\log_k 202 - \lambda \log_k 2}{\lambda - 1}$.

By equating these two equations, we get \[ \frac{\log_k 22 - \lambda \log_k 20}{\lambda - 1} = \frac{\log_k 202 - \lambda \log_k 2}{\lambda - 1} . \]

Therefore, \begin{align*} \log_{20x} (22x) & = \lambda \\ & = \frac{\log_k 22 - \log_k 202}{\log_k 20 - \log_k 2} \\ & = \frac{\log_k \frac{11}{101}}{\log_k 10} \\ & = \log_{10} \frac{11}{101} . \end{align*}

Therefore, the answer is $11 + 101 = \boxed{\textbf{(112) }}$.

~Steven Chen (www.professorchenedu.com)

See Also

2022 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 3 		Followed by
Problem 5
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All AIME Problems and Solutions

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