2021 AMC 12B Problems/Problem 21
Contents
Problem
Let 
be the sum of all positive real numbers 
for which![\[x^{2^{\sqrt2}}=\sqrt2^{2^x}.\]](http://latex.artofproblemsolving.com/9/1/d/91d43f026978eb57bfbe0d83263ca432924008bf.png)
Which of the following statements is true?
Video Solution by OmegaLearn (Logarithmic Tricks)
~ pi_is_3.14
Solution (Rough Approximation)
Note that this solution is not recommended.
Upon pure observation, it is obvious that one solution to this equality is 
. From this, we can deduce that this equality has two solutions, since 
grows faster than $x^(2^\sqrt{2})$ (Error compiling LaTeX. Unknown error_msg) (for greater values of ) and is greater than $x^(2^\sqrt{2})$ (Error compiling LaTeX. Unknown error_msg) for 
and less than $x^(2^\sqrt{2})$ (Error compiling LaTeX. Unknown error_msg) for 
, where 
is the second solution. Thus, the answer cannot be 
or 
. We then start plugging in numbers to roughly approximate the answer. When 
, $x^(2^\sqrt{2})>\sqrt{2}^(2^x)$ (Error compiling LaTeX. Unknown error_msg), thus the answer cannot be 
. Then, when 
, $x^(2^\sqrt{2})=4^(2^\sqrt{2})<64<\sqrt{2}^(2^x)=256$ (Error compiling LaTeX. Unknown error_msg). Therefore, 
, so the answer is 
.
See Also
| 2021 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 20 |
Followed by Problem 22 |
| 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 12 Problems and Solutions | |
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