Difference between revisions of "2005 AIME II Problems/Problem 5"

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(Solution)
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Note that Inclusion-Exclusion does not need to be used, as the problem is asking for ordered pairs <math>(a,b)</math>, and not for the number of possible values of <math>b</math>.  Were the problem to ask for the number of possible values of <math>b</math>, the values of <math>b^6</math> under <math>2005</math> would have to be subtracted, which would just be <math>2</math> values:  <math>2^6</math> and <math>3^6</math>.
 
Note that Inclusion-Exclusion does not need to be used, as the problem is asking for ordered pairs <math>(a,b)</math>, and not for the number of possible values of <math>b</math>.  Were the problem to ask for the number of possible values of <math>b</math>, the values of <math>b^6</math> under <math>2005</math> would have to be subtracted, which would just be <math>2</math> values:  <math>2^6</math> and <math>3^6</math>.
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==Solution II ==
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Let <math>a=log_a b</math>. Then our equation becomes <math>a+\frac{1}{a}=5</math>. Multiplying through by <math>a</math> and solving the quadratic gives us <math>a=2</math> or <math>a=3</math>. Hence <math>a^2=b</math> or <math>a^3=b</math>. 
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For the first case <math>a^2=b</math>, <math>a</math> can range from 2 to 44, a total of 43 values.
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For the second case <math>a^3=b</math>, <math>a</math> can range from 2 to 12, a total of 11 values.
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Thus the total number of possible values is <math>43+11=\boxed{54}</math>.
  
 
== See also ==
 
== See also ==

Revision as of 01:13, 1 March 2011

Problem

Determine the number of ordered pairs $(a,b)$ of integers such that $\log_a b + 6\log_b a=5, 2 \leq a \leq 2005,$ and $2 \leq b \leq 2005.$

Solution

The equation can be rewritten as $\frac{\log b}{\log a} + 6 \frac{\log a}{\log b} = \frac{(\log b)^2+6(\log a)^2}{\log a  \log b}=5$ Multiplying through by $\log a \log b$ and factoring yields $(\log b - 3\log a)(\log b - 2\log a)=0$. Therefore, $\log b=3\log a$ or $\log b=2\log a$, so either $b=a^3$ or $b=a^2$.

  • For the case $b=a^2$, note that $44^2=1936$ and $45^2=2025$. Thus, all values of $a$ from $2$ to $44$ will work.
  • For the case $b=a^3$, note that $12^3=1728$ while $13^3=2197$. Therefore, for this case, all values of $a$ from $2$ to $12$ work.

There are $44-2+1=43$ possibilities for the square case and $12-2+1=11$ possibilities for the cube case. Thus, the answer is $43+11= \boxed{054}$.

Note that Inclusion-Exclusion does not need to be used, as the problem is asking for ordered pairs $(a,b)$, and not for the number of possible values of $b$. Were the problem to ask for the number of possible values of $b$, the values of $b^6$ under $2005$ would have to be subtracted, which would just be $2$ values: $2^6$ and $3^6$.


Solution II

Let $a=log_a b$. Then our equation becomes $a+\frac{1}{a}=5$. Multiplying through by $a$ and solving the quadratic gives us $a=2$ or $a=3$. Hence $a^2=b$ or $a^3=b$.

For the first case $a^2=b$, $a$ can range from 2 to 44, a total of 43 values. For the second case $a^3=b$, $a$ can range from 2 to 12, a total of 11 values.


Thus the total number of possible values is $43+11=\boxed{54}$.

See also

2005 AIME II (ProblemsAnswer KeyResources)
Preceded by
Problem 4
Followed by
Problem 6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions
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