Difference between revisions of "2023 AIME I Problems/Problem 2"
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==Problem== | ==Problem== | ||
− | + | Positive real numbers <math>b \not= 1</math> and <math>n</math> satisfy the equations <cmath>\sqrt{\log_b n} = \log_b \sqrt{n} \qquad \text{and} \qquad b \cdot \log_b n = \log_b (bn).</cmath> The value of <math>n</math> is <math>\frac{j}{k},</math> where <math>j</math> and <math>k</math> are relatively prime positive integers. Find <math>j+k.</math> | |
− | + | ==Solution 1== | |
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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 27: | Line 25: | ||
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ||
− | + | ==Solution 2== | |
Solution by someone else | Solution by someone else | ||
==See also== | ==See also== | ||
{{AIME box|year=2023|num-b=1|num-a=3|n=I}} | {{AIME box|year=2023|num-b=1|num-a=3|n=I}} |
Revision as of 13:24, 8 February 2023
Contents
[hide]Problem
Positive real numbers and satisfy the equations The value of is where and are relatively prime positive integers. Find
Solution 1
Denote . Hence, the system of equations given in the problem can be rewritten as
Thus, and . Therefore,
Therefore, the answer is .
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Solution 2
Solution by someone else
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 |