MathDB

1

Part of 2017 Vietnam National Olympiad

Problems(2)

Convergent sequence

Source: 2017 VMO Problem 1

1/11/2017
Given a∈Ra\in\mathbb{R} and a sequence (un)(u_n) defined by {u1=aun+1=12+2n+3n+1un+14emsp;∀n∈N∗ \begin{cases} u_1=a\\ u_{n+1}=\frac{1}{2}+\sqrt{\frac{2n+3}{n+1}u_n+\frac{1}{4}} \forall n\in\mathbb{N}^* \end{cases}
a) Prove that (un)(u_n) is convergent sequence when a=5a=5 and find the limit of the sequence in that case
b) Find all aa such that the sequence (un)(u_n) is exist and is convergent.
algebrareal analysis
Problem 5 of VMO 2017

Source: Viet Nam National Olympiad 2017

1/6/2017
Find all functions f:R→Rf: \mathbb{R} \rightarrow \mathbb{R} satisfying relation : f(xf(y)−f(x))=2f(x)+xyf(xf(y)-f(x))=2f(x)+xy ∀x,y∈R\forall x,y \in \mathbb{R}
functionalgebra