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 () 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
Now, for all , in (*) setting we have
Final,from (3),(4) and idea of Example 1.10 () we see that , but
, we have
Link download http://tuan.nguyentrung.googlepages.com/A_famous_functional_equation.pdf
 Functional equations and How to solve them, Christopher G.Small.