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cos^{100} x + a_1 cos^{99} x + a_2cos^{98} x +... + a_99 cos x+ a_{100} = 0

Source: : IV Soros Olympiad 1997-98 R3 11.8 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

June 2, 2024
trigonometryalgebrapolynomial

Problem Statement

Sum of all roots of the equation cos100x+a1cos99x+a2cos98x+...+a99cosx+a100=0cos^{100} x + a_1 cos^{99} x + a_2cos^{98} x +... + a_99 cos x+ a_{100} = 0, in interval [π,3π2]\left[\pi, \frac{3\pi}{2} \right], is equal to 21π21\pi, and the sum of all roots of the equation sin100x+a1sin99x+a2sin98x+...+a99sinx+a100=0sin^{100} x + a_1 sin^{99} x + a_2sin ^{98} x +... + a_99sin x+ a_{100} = 0, in the same interval, is equal to 24π24\pi . How many roots does the first equation have on the segment [π2,π]\left[ \frac{\pi}{2}, \pi\right]?
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