Difference between revisions of "2024 AIME I Problems/Problem 2"
Line 21: | Line 21: | ||
Convert the two equations into exponents: | Convert the two equations into exponents: | ||
+ | |||
<cmath>x^{10}=y^x~(1)</cmath> | <cmath>x^{10}=y^x~(1)</cmath> | ||
<cmath>y^{10}=x^{4y}~(2).</cmath> | <cmath>y^{10}=x^{4y}~(2).</cmath> | ||
Line 27: | Line 28: | ||
<cmath>x^{\frac{10}{y}}=y.</cmath> | <cmath>x^{\frac{10}{y}}=y.</cmath> | ||
− | Plug this into | + | |
+ | Plug this into (2): | ||
+ | |||
+ | <math></math>x^({\frac{10}{y}})(10)=x^{4x^{\frac{10}{y}} | ||
Revision as of 19:54, 2 February 2024
Problem
There exist real numbers and
, both greater than 1, such that
. Find
.
Solution 1
By properties of logarithms, we can simplify the given equation to . Let us break this into two separate equations:
\begin{align*}
x\log_xy&=10 \\
4y\log_yx&=10. \\
\end{align*}
We multiply the two equations to get:
Also by properties of logarithms, we know that ; thus,
. Therefore, our equation simplifies to:
~Technodoggo
Solution 2 (if you're bad at logs)
Convert the two equations into exponents:
Take to the power of
:
Plug this into (2):
$$ (Error compiling LaTeX. Unknown error_msg)x^({\frac{10}{y}})(10)=x^{4x^{\frac{10}{y}}
See also
2024 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.