Difference between revisions of "2023 AIME I Problems/Problem 2"
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− | + | Solving our system gives <math>x = 4</math> and <math>b = \frac{5}{4}</math>. | |
Therefore, | Therefore, | ||
<cmath>n = b^x = \frac{625}{256}.</cmath> | <cmath>n = b^x = \frac{625}{256}.</cmath> |
Revision as of 13:35, 9 February 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 our system gives
and
.
Therefore,
Therefore, the answer is
.
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
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.