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Real Solutions to Trig Equation

Source: 2014 AMC 12B #25

February 20, 2014
trigonometryAMC

Problem Statement

What is the sum of all positive real solutions xx to the equation 2cos⁡(2x)(cos⁡(2x)−cos⁡(2014π2x))=cos⁡(4x)−1?2\cos(2x)\left(\cos(2x)-\cos\left(\frac{2014\pi^2}{x}\right)\right)=\cos(4x)-1? <spanclass=′latex−bold′>(A)</span>π<spanclass=′latex−bold′>(B)</span>810π<spanclass=′latex−bold′>(C)</span>1008π<spanclass=′latex−bold′>(D)</span>1080π<spanclass=′latex−bold′>(E)</span>1800π<span class='latex-bold'>(A) </span>\pi\qquad <span class='latex-bold'>(B) </span>810\pi\qquad <span class='latex-bold'>(C) </span>1008\pi\qquad <span class='latex-bold'>(D) </span>1080\pi\qquad <span class='latex-bold'>(E) </span>1800\pi\qquad
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