MathDB

2

Part of 1987 Vietnam National Olympiad

Problems(2)

two sequences

Source: Vietnam NMO 1987 Problem 2

2/4/2009
Sequences (xn) (x_n) and (yn) (y_n) are constructed as follows: x_0 \equal{} 365, x_{n\plus{}1} \equal{} x_n\left(x^{1986} \plus{} 1\right) \plus{} 1622, and y_0 \equal{} 16, y_{n\plus{}1} \equal{} y_n\left(y^3 \plus{} 1\right) \minus{} 1952, for all n≥0 n \ge 0. Prove that \left|x_n\minus{} y_k\right|\neq 0 for any positive integers n n, k k.
algorithmalgebrapolynomialmodular arithmeticalgebra unsolved
Differentiable function

Source: Vietnam NMO 1987 Problem 5

2/4/2009
Let f : [0, \plus{}\infty) \to \mathbb R be a differentiable function. Suppose that ∣f(x)∣≤5 \left|f(x)\right| \le 5 and f(x)f′(x)≥sin⁡x f(x)f'(x) \ge \sin x for all x≥0 x \ge 0. Prove that there exists \lim_{x\to\plus{}\infty}f(x).
functiontrigonometrylimitintegrationalgebra unsolvedalgebra