# IMO 2007 Short list problem

Find all functions  $\large f:\mathbb{R+} \rightarrow \mathbb{R+}$ such that

for all x,y $\large \in$  $\large \mathbb{R+}$

Solution :

For any positive real numbers , we have that

and by Cauchy in positive reals, then $f(x)=\alpha x$ for all $x\in (0,\infty)$
Now it’s easy to see that $\alpha=2$ , then $f(x)=2x$    for all positive real numbers $x$

I am a young student who loves math . I like number theory and inequalities part the most , and preparing for Math Olympiads :)

Posted on Friday,July 6, 2012, in Functional equations, Problems and tagged , , , . Bookmark the permalink. 7 Comments.

1. ricky

good question !
f(x+f(y)) = f(x+y)+f(y)
because f:R+ -> R+
there is function : f(0) = p and f(q) = 0
first substitute : x=0
f(f(y)) = f(y)+f(y)
f(f(y)) = 2f(y)
let u=f(y)
f(u) = 2u
then f(x) = 2x 🙂

2. keshushivang

Read the question carefully !! 😀
its given that function is only for positive Real numbers
so x can’t have value as 0 😀

3. Tom Sawyer

f(x)=2x

• keshushivang