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Vietnam NMO 2000_4

Source:

October 26, 2008
trigonometryalgebrapolynomialalgebra unsolved

Problem Statement

For every integer n≥3 n \ge 3 and any given angle α \alpha with 0<α<π 0 < \alpha < \pi, let P_n(x) \equal{} x^n \sin\alpha \minus{} x \sin n\alpha \plus{} \sin(n \minus{} 1)\alpha. (a) Prove that there is a unique polynomial of the form f(x) \equal{} x^2 \plus{} ax \plus{} b which divides Pn(x) P_n(x) for every n≥3 n \ge 3. (b) Prove that there is no polynomial g(x) \equal{} x \plus{} c which divides Pn(x) P_n(x) for every n≥3 n \ge 3.
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