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Difference between revisions of "1994 AHSME Problems/Problem 3"

(Created page with "==Problem== How many of the following are equal to <math>x^x+x^x</math> for all <math>x>0</math>? <math>\textbf{I:}\ 2x^x \qquad\textbf{II:}\ x^{2x} \qquad\textbf{III:}\ (2x)^x ...")
 
(→‎Solution)
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<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4 </math>
 
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4 </math>
 
==Solution==
 
==Solution==
 +
We look at each statement individually.
 +
 +
<math>\textbf{I:}\ 2x^x</math>. We note that <math>x^x+x^x=x^x(1+1)=2x^x</math>. So statement <math>\textbf{I}</math> is true.
 +
 +
<math>\textbf{II:}\ x^{2x}</math>. We find a counter example which is <math>x=1</math>. <math>2\neq 1</math>. So statement <math>\textbf{II}</math> is false.
 +
 +
<math>\textbf{III:}\ (2x)^x</math>. We see that this statement is equal to <math>2^xx^x</math>. <math>x=2</math> is a counter example. <math>8\neq 16</math>. So statement <math>\textbf{III}</math> is false.
 +
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<math>\textbf{IV:}\ (2x)^{2x}</math>. We see that <math>x=1</math> is again a counter example. <math>2\neq 4</math>. So statement <math>\textbf{IV}</math> is false.
 +
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Therefore, our answer is <math>\boxed{\textbf{(B) }1}</math>.
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 +
--Solution by [http://www.artofproblemsolving.com/Forum/memberlist.php?mode=viewprofile&u=200685 TheMaskedMagician]

Revision as of 15:06, 28 June 2014

Problem

How many of the following are equal to $x^x+x^x$ for all $x>0$?

$\textbf{I:}\ 2x^x \qquad\textbf{II:}\ x^{2x} \qquad\textbf{III:}\ (2x)^x \qquad\textbf{IV:}\ (2x)^{2x}$

$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$

Solution

We look at each statement individually.

$\textbf{I:}\ 2x^x$. We note that $x^x+x^x=x^x(1+1)=2x^x$. So statement $\textbf{I}$ is true.

$\textbf{II:}\ x^{2x}$. We find a counter example which is $x=1$. $2\neq 1$. So statement $\textbf{II}$ is false.

$\textbf{III:}\ (2x)^x$. We see that this statement is equal to $2^xx^x$. $x=2$ is a counter example. $8\neq 16$. So statement $\textbf{III}$ is false.

$\textbf{IV:}\ (2x)^{2x}$. We see that $x=1$ is again a counter example. $2\neq 4$. So statement $\textbf{IV}$ is false.

Therefore, our answer is $\boxed{\textbf{(B) }1}$.

--Solution by TheMaskedMagician

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