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Turkish NMO First Round - 2012 Problem - 27 {Algebra}

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July 1, 2012
trigonometry

Problem Statement

What is the least real number CC that satisfies sin⁡xcos⁡x≤C(sin⁡6x+cos⁡6x)\sin x \cos x \leq C(\sin^6x+\cos^6x) for every real number xx?
<spanclass=′latex−bold′>(A)</span> 3<spanclass=′latex−bold′>(B)</span> 22<spanclass=′latex−bold′>(C)</span> 2<spanclass=′latex−bold′>(D)</span> 2<spanclass=′latex−bold′>(E)</span> None <span class='latex-bold'>(A)</span>\ \sqrt3 \qquad <span class='latex-bold'>(B)</span>\ 2\sqrt2 \qquad <span class='latex-bold'>(C)</span>\ \sqrt 2 \qquad <span class='latex-bold'>(D)</span>\ 2 \qquad <span class='latex-bold'>(E)</span>\ \text{None}
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