Difference between revisions of "2022 AIME II Problems/Problem 4"
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==Video Solution== | ==Video Solution== | ||
https://www.youtube.com/watch?v=4qJyvyZN630 | https://www.youtube.com/watch?v=4qJyvyZN630 | ||
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+ | ==Video Solution by Power of Logic== | ||
+ | https://youtu.be/8eb0ycrVWIM | ||
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+ | ~Hayabusa1 | ||
==See Also== | ==See Also== | ||
{{AIME box|year=2022|n=II|num-b=3|num-a=5}} | {{AIME box|year=2022|n=II|num-b=3|num-a=5}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 16:15, 6 March 2022
Contents
Problem
There is a positive real number not equal to either
or
such that
The value
can be written as
, where
and
are relatively prime positive integers. Find
.
Solution 1
Define to be
, what we are looking for. Then, by the definition of logs,
Dividing the first equation by the second equation gives us
, so by the definition of logs,
. This is what the problem asked for, so the fraction
gives us
.
~ihatemath123
Solution 2
We could assume a variable which equals to both
and
.
So that
and
Express as:
Substitute to
:
Thus, , where
and
.
Therefore, .
~DSAERF-CALMIT (https://binaryphi.site)
Solution 3
We have
We have
Because , we get
We denote this common value as .
By solving the equality , we get
.
By solving the equality , we get
.
By equating these two equations, we get
Therefore,
Therefore, the answer is .
~Steven Chen (www.professorchenedu.com)
Solution 4
By the change of base rule, we have , or
. We also know that if
, then this also equals
. We use this identity and find that
. The requested sum is
~MathIsFun286
Video Solution
https://www.youtube.com/watch?v=4qJyvyZN630
Video Solution by Power of Logic
~Hayabusa1
See Also
2022 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
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.