Difference between revisions of "2005 AIME II Problems/Problem 5"
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Determine the number of [[ordered pair]]s <math> (a,b) </math> of [[integer]]s such that <math> \log_a b + 6\log_b a=5, 2 \leq a \leq 2005, </math> and <math> 2 \leq b \leq 2005. </math> | Determine the number of [[ordered pair]]s <math> (a,b) </math> of [[integer]]s such that <math> \log_a b + 6\log_b a=5, 2 \leq a \leq 2005, </math> and <math> 2 \leq b \leq 2005. </math> | ||
− | == Solution == | + | == Solution 1 == |
The equation can be rewritten as <math> \frac{\log b}{\log a} + 6 \frac{\log a}{\log b} = \frac{(\log b)^2+6(\log a)^2}{\log a \log b}=5 </math> Multiplying through by <math>\log a \log b </math> and factoring yields <math>(\log b - 3\log a)(\log b - 2\log a)=0 </math>. Therefore, <math>\log b=3\log a </math> or <math>\log b=2\log a </math>, so either <math> b=a^3 </math> or <math> b=a^2 </math>. | The equation can be rewritten as <math> \frac{\log b}{\log a} + 6 \frac{\log a}{\log b} = \frac{(\log b)^2+6(\log a)^2}{\log a \log b}=5 </math> Multiplying through by <math>\log a \log b </math> and factoring yields <math>(\log b - 3\log a)(\log b - 2\log a)=0 </math>. Therefore, <math>\log b=3\log a </math> or <math>\log b=2\log a </math>, so either <math> b=a^3 </math> or <math> b=a^2 </math>. | ||
*For the case <math> b=a^2 </math>, note that <math> 44^2=1936 </math> and <math> 45^2=2025 </math>. Thus, all values of <math>a</math> from <math>2</math> to <math>44</math> will work. | *For the case <math> b=a^2 </math>, note that <math> 44^2=1936 </math> and <math> 45^2=2025 </math>. Thus, all values of <math>a</math> from <math>2</math> to <math>44</math> will work. | ||
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There are <math> 44-2+1=43 </math> possibilities for the square case and <math> 12-2+1=11 </math> possibilities for the cube case. Thus, the answer is <math> 43+11= \boxed{054}</math>. | There are <math> 44-2+1=43 </math> possibilities for the square case and <math> 12-2+1=11 </math> possibilities for the cube case. Thus, the answer is <math> 43+11= \boxed{054}</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>. | + | 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>. However, the ordered pairs where b is to the sixth power are distinct, so they are not redundant. (For example, the pairs (4, 64) and (8, 64).) |
− | + | ==Solution 2 == | |
− | + | Let <math>k=\log_a b</math>. Then our equation becomes <math>k+\frac{6}{k}=5</math>. Multiplying through by <math>k</math> and solving the quadratic gives us <math>k=2</math> or <math>k=3</math>. Hence <math>a^2=b</math> or <math>a^3=b</math>. | |
− | ==Solution | ||
− | Let <math> | ||
For the first case <math>a^2=b</math>, <math>a</math> can range from 2 to 44, a total of 43 values. | 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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[[Category:Intermediate Algebra Problems]] | [[Category:Intermediate Algebra Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 14:27, 27 November 2019
Contents
Problem
Determine the number of ordered pairs of integers such that and
Solution 1
The equation can be rewritten as Multiplying through by and factoring yields . Therefore, or , so either or .
- For the case , note that and . Thus, all values of from to will work.
- For the case , note that while . Therefore, for this case, all values of from to work.
There are possibilities for the square case and possibilities for the cube case. Thus, the answer is .
Note that Inclusion-Exclusion does not need to be used, as the problem is asking for ordered pairs , and not for the number of possible values of . Were the problem to ask for the number of possible values of , the values of under would have to be subtracted, which would just be values: and . However, the ordered pairs where b is to the sixth power are distinct, so they are not redundant. (For example, the pairs (4, 64) and (8, 64).)
Solution 2
Let . Then our equation becomes . Multiplying through by and solving the quadratic gives us or . Hence or .
For the first case , can range from 2 to 44, a total of 43 values. For the second case , can range from 2 to 12, a total of 11 values.
Thus the total number of possible values is .
See also
2005 AIME II (Problems • Answer Key • Resources) | ||
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.