Difference between revisions of "2021 AMC 12B Problems/Problem 21"

(Solution (Rough Approximation))
(Solution (Rough Approximation))
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== Solution (Rough Approximation) ==
 
== Solution (Rough Approximation) ==
Note that this solution is not recommended.
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Note that this solution is not recommended unless you're running out of time.
  
Upon pure observation, it is obvious that one solution to this equality is <math>x=\sqrt{2}</math>. From this, we can deduce that this equality has two solutions, since <math>\sqrt{2}^(2^x)</math> grows faster than <math>x^(2^\sqrt{2})</math> (for greater values of <math>x</math>) and <math>\sqrt{2}^(2^x)</math> is greater than <math>x^(2^\sqrt{2})</math> for <math>x<\sqrt{2}</math> and less than <math>x^(2^\sqrt{2})</math> for <math>\sqrt{2}<x<n</math>, where <math>n</math> is the second solution. Thus, the answer cannot be <math>A</math> or <math>B</math>. We then start plugging in numbers to roughly approximate the answer. When <math>x=2</math>, <math>x^(2^\sqrt{2})>\sqrt{2}^(2^x)</math>, thus the answer cannot be <math>C</math>. Then, when <math>x=4</math>, <math>x^(2^\sqrt{2})=4^(2^\sqrt{2})<64<\sqrt{2}^(2^x)=256</math>. Therefore, <math>S<4+\sqrt{2}<6</math>, so the answer is <math>\boxed{D}</math>.
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Upon pure observation, it is obvious that one solution to this equality is <math>x=\sqrt{2}</math>. From this, we can deduce that this equality has two solutions, since <math>\sqrt{2}^{2^{x}}</math> grows faster than <math>x^{2^{\sqrt{2}}}</math> (for greater values of <math>x</math>) and <math>\sqrt{2}^{2^{x}}</math> is greater than <math>x^{2^{\sqrt{2}}}</math> for <math>x<\sqrt{2}</math> and less than <math>x^{2^{\sqrt{2}}}</math> for <math>\sqrt{2}<x<n</math>, where <math>n</math> is the second solution. Thus, the answer cannot be <math>A</math> or <math>B</math>. We then start plugging in numbers to roughly approximate the answer. When <math>x=2</math>, <math>x^{2^{\sqrt{2}}}>\sqrt{2}^{2^{x}}</math>, thus the answer cannot be <math>C</math>. Then, when <math>x=4</math>, <math>x^{2^{\sqrt{2}}}=4^{2^{\sqrt{2}}}<64<\sqrt{2}^{2^{x}}=256</math>. Therefore, <math>S<4+\sqrt{2}<6</math>, so the answer is <math>\boxed{D}</math>.
  
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2021|ab=B|num-b=20|num-a=22}}
 
{{AMC12 box|year=2021|ab=B|num-b=20|num-a=22}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 01:20, 12 February 2021

Problem

Let $S$ be the sum of all positive real numbers $x$ for which\[x^{2^{\sqrt2}}=\sqrt2^{2^x}.\]Which of the following statements is true?

$\textbf{(A) }S<\sqrt2 \qquad \textbf{(B) }S=\sqrt2 \qquad \textbf{(C) }\sqrt2<S<2\qquad \textbf{(D) }2\le S<6 \qquad \textbf{(E) }S\ge 6$

Video Solution by OmegaLearn (Logarithmic Tricks)

https://youtu.be/uCTpLB-kGR4

~ pi_is_3.14

Solution (Rough Approximation)

Note that this solution is not recommended unless you're running out of time.

Upon pure observation, it is obvious that one solution to this equality is $x=\sqrt{2}$. From this, we can deduce that this equality has two solutions, since $\sqrt{2}^{2^{x}}$ grows faster than $x^{2^{\sqrt{2}}}$ (for greater values of $x$) and $\sqrt{2}^{2^{x}}$ is greater than $x^{2^{\sqrt{2}}}$ for $x<\sqrt{2}$ and less than $x^{2^{\sqrt{2}}}$ for $\sqrt{2}<x<n$, where $n$ is the second solution. Thus, the answer cannot be $A$ or $B$. We then start plugging in numbers to roughly approximate the answer. When $x=2$, $x^{2^{\sqrt{2}}}>\sqrt{2}^{2^{x}}$, thus the answer cannot be $C$. Then, when $x=4$, $x^{2^{\sqrt{2}}}=4^{2^{\sqrt{2}}}<64<\sqrt{2}^{2^{x}}=256$. Therefore, $S<4+\sqrt{2}<6$, so the answer is $\boxed{D}$.

See Also

2021 AMC 12B (ProblemsAnswer KeyResources)
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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