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P (cos x/ 1999t) = 1/2 (V Soros Olympiad 1998-99 Round 2 10.3)

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May 25, 2024
algebratrigonometry

Problem Statement

It is known that sin⁡3x=3sin⁡x−4sin⁡3x\sin 3x = 3 \sin x - 4 \sin^3x. It is also easy to prove that sin⁡nx\sin nx for odd nn can be represented as a polynomial of degree nn of sin⁡x\sin x. Let sin⁡1999x=P(sin⁡x)\sin 1999x = P(\sin x), where P(t)P(t) is a polynomial of the 19991999th degree of tt. Solve the equation P(cos⁡x1999)=12.P \left(\cos \frac{x}{1999}\right) = \frac12 .
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