1994 AHSME Problems/Problem 3

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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1994 AHSME Problems/Problem 3

Problem

Solution