2023 AIME I Problems/Problem 2
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 (extremely similar to above)
First, take the first equation and convert to . Square both sides to get . Because a logarithm cannot be equal to , .
By another logarithm rule, . Therefore, , and . Since , we have , and .
~wuwang2002 (feel free to remove if this is too similar to the above)
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.