Difference between revisions of "2019 AMC 12A Problems/Problem 12"
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We therefore have <math>k+\frac{4}{k}=6</math>, and deduce <math>k^2-6k+4=0</math>. The solutions to this are <math>k = 3 \pm \sqrt{5}</math>. | We therefore have <math>k+\frac{4}{k}=6</math>, and deduce <math>k^2-6k+4=0</math>. The solutions to this are <math>k = 3 \pm \sqrt{5}</math>. | ||
− | To solve the problem, we now find < | + | To solve the problem, we now find |
+ | <cmath>\begin{align*} | ||
+ | (\log_2\tfrac{x}{y})^2&=(\log_2 x - \log_2 y)^2\ | ||
+ | &=(k-\tfrac{4}{k})^2=(3 \pm \sqrt{5} - \tfrac{4}{3 \pm \sqrt{5}})^2 \ | ||
+ | &= (3 \pm \sqrt{5} - [3 \mp \sqrt{5}])^2\ | ||
+ | &= (3 \pm \sqrt{5} - 3 \pm \sqrt{5})^2\ | ||
+ | &=(\pm 2\sqrt{5})^2 \ | ||
+ | &= \boxed{\textbf{(B) } 20}. \ | ||
+ | \end{align*}</cmath> | ||
+ | ~Edits by BakedPotato66 | ||
==Solution 2 (slightly simpler)== | ==Solution 2 (slightly simpler)== | ||
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We know that <math>xy=64</math>, so <math>x= \frac{64}{y}</math>. | We know that <math>xy=64</math>, so <math>x= \frac{64}{y}</math>. | ||
− | Thus <math>\log_2(\frac{64}{y}) = \frac{4}{{\log_{2}(y)}}</math>, so <math>6-\log_2(y) = \frac{4}{{\log_{2}(y)}}</math>, so <math>6(\log_2(y))-(\log_2(y))^2=4</math>. | + | Thus <math>\log_2\left(\frac{64}{y}\right) = \frac{4}{{\log_{2}(y)}}</math>, so <math>6-\log_2(y) = \frac{4}{{\log_{2}(y)}}</math>, so <math>6(\log_2(y))-(\log_2(y))^2=4</math>. |
Solving for <math>\log_2(y)</math>, we obtain <math>\log_2(y)=3+\sqrt{5}</math>. | Solving for <math>\log_2(y)</math>, we obtain <math>\log_2(y)=3+\sqrt{5}</math>. | ||
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Our goal is to find <math>( A-B )^2</math>. From the above, it is equal to <math>(6-2B) = \left(2\sqrt{5}\right)^2 = 20 \Rightarrow \boxed{B}</math>. | Our goal is to find <math>( A-B )^2</math>. From the above, it is equal to <math>(6-2B) = \left(2\sqrt{5}\right)^2 = 20 \Rightarrow \boxed{B}</math>. | ||
− | == Video Solution == | + | Alternatively, once we found <math>AB=4</math> and <math>A+B=6</math>, we could have squared the latter to get <math>A^2+B^2+2AB=36</math>; subtracting <math>4</math> times the former equation, we find that <math>A^2+B^2-2AB=(A-B)^2=36-16=\boxed{\textbf{(B) }20}</math>. (Alternate finish by Technodoggo) |
+ | |||
+ | ==Video Solution 1== | ||
+ | https://youtu.be/ODOWgzhVKog | ||
+ | |||
+ | ~Education, the Study of Everything | ||
+ | |||
+ | == Video Solution by OmegaLearn == | ||
https://youtu.be/RdIIEhsbZKw?t=1821 | https://youtu.be/RdIIEhsbZKw?t=1821 | ||
Latest revision as of 00:16, 3 November 2024
Contents
[hide]Problem
Positive real numbers and satisfy and . What is ?
Solution 1
Let , so that and . Then we have .
We therefore have , and deduce . The solutions to this are .
To solve the problem, we now find ~Edits by BakedPotato66
Solution 2 (slightly simpler)
After obtaining , notice that the required answer is , as before.
Solution 3
From the given data, , or
We know that , so .
Thus , so , so .
Solving for , we obtain .
Easy resubstitution further gives . Simplifying, we obtain .
Looking back at the original problem, we have What is ?
Deconstructing this expression using log rules, we get .
Plugging in our known values, we get or .
Our answer is .
Solution 4
Multiplying the first equation by , we obtain .
From the second equation we have .
Then, .
Solution 5
Let and .
Writing the first given as and the second as , we get and .
Solving for we get .
Our goal is to find . From the above, it is equal to .
Alternatively, once we found and , we could have squared the latter to get ; subtracting times the former equation, we find that . (Alternate finish by Technodoggo)
Video Solution 1
~Education, the Study of Everything
Video Solution by OmegaLearn
https://youtu.be/RdIIEhsbZKw?t=1821
~ pi_is_3.14
See Also
2019 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 11 |
Followed by Problem 13 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.