# Difference between revisions of "2007 BMO Problems/Problem 2"

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\displaystyle f(f(x) + y) = f(f(x) - y) + 4f(x)y | \displaystyle f(f(x) + y) = f(f(x) - y) + 4f(x)y | ||

− | </math>. | + | </math>, for any <math> x,y \in \mathbb{R} </math>. |

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## Latest revision as of 22:46, 4 May 2007

## Problem

(*Bulgaria*)
Find all functions such that

, for any .

## Solution

We first note that is a solution to the equation. Henceforth we shall consider other solutions to the equations, i.e., functions such that for some , .

Setting gives us .

We note that for any , , i.e., as and assume all real values, assume all real values.

Now, setting and , we obtain

,

or

.

Since takes on all real values, it follows that for all , . It is easy to see that any value of will satisfy the desired condition. Thus the only solutions to the functional equation are and , an arbitrary constant.

*Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.*