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11.8

Part of IV Soros Olympiad 1997 - 98 (Russia)

Problems(2)

approx. \sqrt{5,44...4} (IV Soros Olympiad 1997-98 Correspondence 11.8)

Source:

6/1/2024
Calculate 5,44...4\sqrt{5,44...4} (the decimal point is followed by 100100 fours) with approximation to: a) 10−10010^{-100}, b) 10−20010^{-200}
algebraRadicals
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

6/2/2024
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]?
trigonometryalgebrapolynomial