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

Latest revision as of 16:26, 9 January 2021

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

1994 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
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All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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 --Solution by [http://www.artofproblemsolving.com/Forum/memberlist.php?mode=viewprofile&u=200685 TheMaskedMagician]
+==See Also==
+{{AHSME box|year=1994|num-b=2|num-a=4}}
+{{MAA Notice}}

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

Latest revision as of 16:26, 9 January 2021

Problem

Solution

See Also