MathDB

1

Part of 2000 Vietnam National Olympiad

Problems(2)

Vietnam NMO 2000_1

Source:

10/26/2008
Given a real number c>0 c > 0, a sequence (xn) (x_n) of real numbers is defined by x_{n \plus{} 1} \equal{} \sqrt {c \minus{} \sqrt {c \plus{} x_n}} for n≥0 n \ge 0. Find all values of c c such that for each initial value x0 x_0 in (0,c) (0, c), the sequence (xn) (x_n) is defined for all n n and has a finite limit lim⁡xn \lim x_n when n\to \plus{} \infty.
limitalgebra unsolvedalgebra
Vietnam NMO 2000_4

Source:

10/26/2008
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.
trigonometryalgebrapolynomialalgebra unsolved