Difference between revisions of "2022 AMC 10A Problems/Problem 11"
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==Solution 3== | ==Solution 3== | ||
− | Since | + | Since third roots are conventionally positive integers, assume <math>m</math> is an integer, so <math>m</math> can only be <math>1</math>, <math>2</math>, <math>3</math>, <math>4</math>, <math>6</math>, and <math>12</math>. <math>\sqrt{\frac{1}{4096}}=\frac{1}{64}</math>. Testing out <math>m</math>, we see that only <math>3</math> and <math>4</math> work. Hence, <math>3+4=\boxed{\textbf{(C) }7}</math>. |
~MrThinker | ~MrThinker |
Latest revision as of 22:35, 10 November 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
and equating exponents, 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 third 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
Video Solution by TheBeautyofMath
~IceMatrix
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.