## Cau truc de thi mon Toan (du kien) nam 2008 cua Bo GD&DT

Dinh lan mot thoi gian, nhung viec nay qua quan trong nen minh da tro lai. 😀 Day la cau truc de thi mon Toan nam 2008,  nhung chi la ”du kien” thoi day! 😛

P.S: May em hoc sinh CHL vao day doc thi nho comment cho thay nha! Thuong cac em nhieu lam. 😀

## 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.