That is following problem: Find all functions such that

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:

Find all real values of , for which there exists one and only one function and satisfying the equation

We easy see that, if is an answer then . In fact,

If . We have at least two functions satisfying, therefore not satisfy.

If . Suppose is function satisfy, because (note that : ) we have is surjection (1).

By (1) exist such that , from (**) we have and , therefore , choose here we have (note that :). So iff (2).

In (**) choose we have (3).

In (3) choose and by (2) we have (4).

In (**) choose we have (5). In (5) choose then . Therefore , so .

So, we’ll back to equation (*). These are solutions of this one:

** 1. Solution of harazi, a member of the site mathlinks.ro.
** In (*) put we have , from here and by above we have

From (1), in (*) Put by we have

From here we have

Now, by and (2) we see that

Final, by (2),(3) and Proposition 2.7 ([1]) we have or . Checking condition (*) we obtain

**2. Solution of N.T.TUAN, a member of the site mathlinks.ro.
**In (*) choose we have .

By above: , if then (because ). Setting and in (*) we have

Therefore , contradiction! So, , form that, see

and .

Now, for all , in (*) setting we have latex f(x)=x\forall x<0$, but

, we have

Link download http://tuan.nguyentrung.googlepages.com/A_famous_functional_equation.pdf

Bibliography

[1] Functional equations and How to solve them, Christopher G.Small.

[2] http://www.mathlinks.ro/viewtopic.php?p=18289

tai sao ơ dòng thư 8 trên xuông lai có f(b)=0 thoa man ca 2 pt

f(x^2+b)= (f(x))^2 +b và …(như trên)

Em nghĩ nó chi thoa man đươc môt pt thôi chư nhi. Tai sao pt dâu trư lai co đươc.