1958 AHSME Problems/Problem 5

Problem

The expression $2 + \sqrt{2} + \frac{1}{2 + \sqrt{2}} + \frac{1}{\sqrt{2} - 2}$ equals:

$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 2 - \sqrt{2}\qquad \textbf{(C)}\ 2 + \sqrt{2}\qquad \textbf{(D)}\ 2\sqrt{2}\qquad \textbf{(E)}\ \frac{\sqrt{2}}{2}$

Solution

We have

\begin{align*}2+\sqrt{2}+\frac{2-\sqrt{2}}{4-2}+\frac{\sqrt{2}+2}{2-4}&=2+\sqrt{2}+\frac{1}{2}(2-\sqrt{2})-\frac{1}{2}(\sqrt{2}+2)\\&=\frac{1}{2}(2+\sqrt{2}+2-\sqrt{2})\\&= \boxed{\text{(A) }2}. (Error compiling LaTeX. Unknown error_msg)

1958 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 4	Followed by Problem 6
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1958 AHSME Problems/Problem 5

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