Difference between revisions of "2022 AMC 10A Problems/Problem 11"
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Revision as of 22:55, 7 September 2023
Contents
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
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 (⚡️lightning fast⚡️)
~Education, the Study of Everything
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.