Difference between revisions of "2024 AIME I Problems/Problem 2"
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~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ||
+ | ==Video Solution== | ||
+ | https://youtu.be/6C0yHp5GUBY | ||
+ | |||
+ | ~Veer Mahajan | ||
==See also== | ==See also== |
Revision as of 18:18, 3 February 2024
Contents
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
Convert the two equations into exponents:
Take to the power of
:
Plug this into :
So
~alexanderruan
Solution 3
Similar to solution 2, we have:
and
Take the tenth root of the first equation to get
Substitute into the second equation to get
This means that , or
, meaning that
.
~MC413551
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution
~Veer Mahajan
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.