Difference between revisions of "2022 AMC 10A Problems/Problem 11"
MRENTHUSIASM (talk | contribs) (Created page with "==Problem== Ted mistakenly wrote <math>2^m\cdot\sqrt{\frac{1}{4096}}</math> as <math>2\cdot\sqrt[m]{\frac{1}{4096}}.</math> What is the sum of all real numbers <math>m</math>...") |
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~MRENTHUSIASM | ~MRENTHUSIASM | ||
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+ | Alternatively, once we reach <math>m-6 = 1-\frac{12}{m}, we rearrange to get </math>m-7+\frac{12}{m}=0<math>. Multiplying both sides by </math>m<math>, we have </math>m^2-7m+12=0<math>. Since were asked to find the sum of all possible values of </math>m<math>, we use vieta’s formula to get the sum of the roots is </math>7=\boxed{C}$ | ||
+ | ~KingRavi | ||
== See Also == | == See Also == |
Revision as of 02:10, 12 November 2022
Problem
Ted mistakenly wrote as
What is the sum of all real numbers
for which these two expressions have the same value?
Solution
We are given that
Converting everything into powers of
we have
We multiply both sides by
, then rearrange and factor as
Therefore, we have
or
The sum of such values of
is
~MRENTHUSIASM
Alternatively, once we reach m-7+\frac{12}{m}=0
m
m^2-7m+12=0
m
7=\boxed{C}$
~KingRavi
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 |
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