Difference between revisions of "2022 AMC 10A Problems/Problem 11"
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from which <math>m = 3</math> or <math>m = 4</math>. Therefore, the answer is <math>3+4 = \boxed{\textbf{(C) } 7}.</math> | from which <math>m = 3</math> or <math>m = 4</math>. Therefore, the answer is <math>3+4 = \boxed{\textbf{(C) } 7}.</math> | ||
− | - | + | - youtube.com/indianmathguy |
==Solution 3== | ==Solution 3== |
Revision as of 15:30, 5 February 2024
Contents
[hide]Problem
Ted mistakenly wrote as
What is the sum of all real numbers
for which these two expressions have the same value?
Solution 1
We are given that
Converting everything into powers of
we have
We multiply both sides by
, then rearrange as
By Vieta's Formulas, the sum of such values of
is
Note that or
from the quadratic equation above.
~MRENTHUSIASM
~KingRavi
Solution 2 (Logarithms)
We can rewrite the equation using fractional exponents and take logarithms of both sides:
We can then use the additive properties of logarithms to split them up:
Using the power rule, the fact that
and bringing the exponents down, we get
from which
or
. Therefore, the answer is
- youtube.com/indianmathguy
Solution 3
Since surd roots are conventionally positive integers, assume is an integer, so
can only be
,
,
,
,
, and
.
. Testing out
, we see that only
and
work. Hence,
.
~MrThinker
Video Solution 1
~Education, the Study of Everything
Video Solution 2
Video Solution 3
~Whiz
See Also
2022 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.