Difference between revisions of "2023 AIME I Problems/Problem 2"
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==Solution2== | ==Solution2== | ||
− | Denote b=x^n | + | Denote b=n^x. Hence, the system of equations given in the problem can be rewritten as |
+ | sqrt(x)=x/2, b*x=1+x | ||
+ | Thus, x=x^2/4, x=4. So, n=b^4 | ||
+ | Then, 4x=1+x. So, x=1/3 | ||
==Video Solution by TheBeautyofMath== | ==Video Solution by TheBeautyofMath== |
Revision as of 08:35, 30 September 2023
Problem
Positive real numbers and satisfy the equations The value of is where and are relatively prime positive integers. Find
Solution
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)
Solution2
Denote b=n^x. Hence, the system of equations given in the problem can be rewritten as sqrt(x)=x/2, b*x=1+x Thus, x=x^2/4, x=4. So, n=b^4 Then, 4x=1+x. So, x=1/3
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.