MathDB

1

Part of 2015 Vietnam National Olympiad

Problems(2)

VMO 2015 Problem 1

Source: VMO 2015

7/30/2016
Given a non negative real aa and a sequence (un)(u_n) defined by {u1=3un+1=un2+n24n2+aun2+3 \begin{cases} u_1=3\\ u_{n+1}=\frac{u_n}{2}+\frac{n^2}{4n^2+a}\sqrt{u_n^2+3} \end{cases}
a) Prove that for a=0a=0, the sequence is convergent and find its limit.
b) For a∈[0,1]a\in [0,1], prove that the sequence if convergent.
algebracalculuslimits
Polynomial

Source: Vietnam National Olympiad 2015 Problem 5

1/11/2015
Let {f(x)}{\left\{ {f(x)} \right\}} be a sequence of polynomial, where f0(x)=2{f_0}(x) = 2, f1(x)=3x{f_1}(x) = 3x, and
fn(x)=3xfn−1(x)+(1−x−2x2)fn−2(x){f_n}(x) = 3x{f_{n - 1}}(x) + (1 - x - 2{x^2}){f_{n - 2}}(x) (n≥2)(n \ge 2)
Determine the value of nn such that fn(x){f_n}(x) is divisible by x3−x2+xx^3-x^2+x.
algebrapolynomialinductionalgebra unsolved