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RU-ALLRUSSIAN 2017 Problem 111

    S=sin⁡64x+sin⁡65xS=\sin{64x}+\sin{65x} and C=cos⁡64x+cos⁡65xC=\cos{64x}+\cos{65x} are both rational for some xx . Prove, that for one of these sums both summands are rational too.

    Dataset text AI-MO AoPS. Quoted with source.

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