2014 AMC 12B Problems/Problem 25

Problem

Find the sum of all the positive solutions of

$2\cos2x \left(\cos2x - \cos{\left( \frac{2014\pi^2}{x} \right) } \right) = \cos4x - 1$

$\textbf{(A)}\ \pi \qquad\textbf{(B)}\ 35\qquad\textbf{(C)}\ 1008\pi \qquad\textbf{(D)}}\ 1080 \pi \qquad\textbf{(E)}\ 1800\pi$ (Error compiling LaTeX. Unknown error_msg)

Solution

Rewrite $\cos{4x} - 1$ as $2\cos^2{2x} - 2$ . Now let $x = \cos{2x}$ , and let $y = \cos{\left( \frac{2014\pi^2}{x} \right) }$ . We have $2x(x - y) = 2x^2 - 2$ $xy = 1$ Notice that $x = 1$ and $y = 1$ or $x = -1$ and $y = -1$ .

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2014 AMC 12B Problems/Problem 25

Problem

Solution