## A famous functional equation

That is following problem: Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
$f(x^2 + y + f(y)) = (f(x))^2 + 2\cdot y\; \forall x,y\in\mathbb{R}.(*)$
It is famous! Why? Because, it is from AMM(problem 10908, posted by Wu Wei Chao) and it is one in problems from Bulgarian TST 2003, Vietnam TST 2004 and Iran TST 2007. However, in Vietnam TST 2004 it has form:

## Problems in Section 1 of GTM 167

Please post carefully solutions of the following ones:

1. Let $K$ be a field extension of $F$. By defining scalar multiplication for $\alpha\in F$ and $a\in K$ by $\alpha\cdot a=\alpha a$, the multiplication in $K$, show that $K$ is an $F-$ vector space.

2. If $K$ is a field extention of $F$, prove that $[K:F]=1$ iff $K=F$.

## Two lemmas on linear sequences of order two

Note: My English is so bad but don’t worry about that! 🙂

A sequence $(a_n)$ is called linear of order two if there are real numbers $p,q$ such that $a_{n+2}=pa_{n+1}+qa_n\;\forall n=1,2,3,...$

In this topic we’ll use ideas in proofs of two following lemmas to solve some Olympiad problems.

## Dãy cho bởi phương trình

Trong de thi HSGQG cua Vietnam chung ta thuong gap bai toan sau: Cho cac phuong trinh $f_n(x) = 0 (n\in\mathbb{N})$ , chung minh moi phuong trinh do co nghiem thuc duy nhat tren mot mien D nao do, goi nghiem do la $x_n$, tim $\lim_{n\to\infty} x_n$.

Topic nay danh de trao doi ve nhung bai toan do, xin moi cac ban post cac loi giai va cung cap cac vi du.