# IMO 2007 Short list problem

Find all functions such that

for all x,y

**Solution :**

For any positive real numbers , we have that

and by Cauchy in positive reals, then for all

Now it’s easy to see that , then for all positive real numbers

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

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 🙂

Read the question carefully !! 😀

its given that function is only for positive Real numbers

so x can’t have value as 0 😀

f(x)=2x

sir give your solution too

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