Difference between revisions of "2023 AIME I Problems/Problem 2"
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https://www.youtube.com/watch?v=jxY7BBe-4gU | https://www.youtube.com/watch?v=jxY7BBe-4gU | ||
− | ==Solution== | + | ==Solution 1== |
Denote <math>x = \log_b n</math>. | Denote <math>x = \log_b n</math>. | ||
Hence, the system of equations given in the problem can be rewritten as | Hence, the system of equations given in the problem can be rewritten as | ||
Line 19: | Line 19: | ||
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ||
+ | |||
+ | ==Solution 2== | ||
+ | We can use the property that <math>\log(xy) = \log(x) + \log(y)</math> on the first equation. We get <math>\log_b(bn) = 1 + \log_b(n)</math>. Then, subtracting <math>\log_b(n)</math> from both sides, we get <math>(b-1) \log_b(n) = 1</math>, therefore <math>\log_b(n) = \frac{1}{b-1}</math>. Substituting that into our first equation, we get <math>\frac{1}{2b-2} = \sqrt{\frac{1}{b-1}}</math>. Squaring, reciprocating, and simplifying both sides, we get the quadratic <math>4b^2 - 9b + 5 = 0</math>. Solving for <math>b</math>, we get <math>\frac{5}{4}</math> and <math>1</math>. Since the problem said that <math>b \neq 1</math>, <math>b = \frac{5}{4}</math>. To solve for <math>n</math>, we can use the property that <math>\log_b(n) = \frac{1}{b-1}</math>. <math>\log_\frac{5}{4}(n) = 4</math>, so <math>n = \frac{5^4}{4^4} = \frac{625}{256}</math>. Adding these together, we get <math>\boxed{881}</math> | ||
+ | |||
+ | ~idk12345678 | ||
+ | |||
+ | ==Solution 3 (quick)== | ||
+ | |||
+ | We can let <math>n=b^{4x^2}</math>. Then, in the first equation, the LHS becomes <math>2x</math> and the RHS becomes <math>2x^2</math>. Therefore, <math>x</math> must be <math>1</math> (<math>x</math> can't be <math>0</math>). So now we know <math>n=b^4</math>. So we can plug this into the second equation to get <math>n=b^4</math>. This gives <math>b\cdot4=5</math>, so <math>b= \frac{5}{4}</math> and <math>n= b^4=\frac{625}{256}</math>. Adding the numerator and denominator gives <math>\boxed{881}</math>. | ||
+ | |||
+ | Honestly this problem is kinda misplaced. | ||
+ | |||
+ | ~yrock | ||
==Video Solution by TheBeautyofMath== | ==Video Solution by TheBeautyofMath== |
Latest revision as of 22:31, 10 September 2024
Contents
[hide]Problem
Positive real numbers and satisfy the equations The value of is where and are relatively prime positive integers. Find
Video Solution & More by MegaMath
https://www.youtube.com/watch?v=jxY7BBe-4gU
Solution 1
Denote . Hence, the system of equations given in the problem can be rewritten as Solving the system gives and . Therefore, Therefore, the answer is .
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Solution 2
We can use the property that on the first equation. We get . Then, subtracting from both sides, we get , therefore . Substituting that into our first equation, we get . Squaring, reciprocating, and simplifying both sides, we get the quadratic . Solving for , we get and . Since the problem said that , . To solve for , we can use the property that . , so . Adding these together, we get
~idk12345678
Solution 3 (quick)
We can let . Then, in the first equation, the LHS becomes and the RHS becomes . Therefore, must be ( can't be ). So now we know . So we can plug this into the second equation to get . This gives , so and . Adding the numerator and denominator gives .
Honestly this problem is kinda misplaced.
~yrock
Video Solution by TheBeautyofMath
~IceMatrix
See also
2023 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
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.