Difference between revisions of "2022 AMC 10A Problems/Problem 11"
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~Education, the Study of Everything | ~Education, the Study of Everything | ||
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== Video Solution (Easy) == | == Video Solution (Easy) == |
Revision as of 19:20, 23 January 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
- abed_nadir
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 (Easy)
~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.