# Internat. Math Olympiad exercise. Weird function.

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1. May 1, 2010

### GBarboza

1. The problem statement, all variables and given/known data

A friend of mine tried to classify for the IMO a few days ago (he didn't do so well). A problem he had to solve was:

f(x + xy + f(y)) = (f(x) + 1/2)(f(x) + 1/2)

I didn't really understand what he said later. First he told me to find the values of X and Y for which this function makes sense, but later I asked him if it said anything more and he told me the domain and codomain were both R, which is a bit weird. As far as I know, the domain is all possible values of X.

2. Relevant equations

I probably misunderstood. I would appreciate it if anyone could tell me what it probably was, and, even better, tell me how to proceed.

3. The attempt at a solution

I got as far as f(x) = -1/2 +- f(x + xy + f(y)) using the quadratic formula... which checks out. My dad says I'm missing information. I don't really understand all this much.

2. May 1, 2010

### Hurkyl

Staff Emeritus
In the problem, you are given an identity satisfied by f, and your goal is to solve for f.

I do believe you have misstated the problem, and suspect you have left out information. (e.g. it is not uncommon to include that f is continuous)

I think you have misstated, because there is a contradiction: if you substitute
x = -f(1)
y = 1​
then you get
f(-f(1)) = (f(-f(1)) + 1/2)(f(-f(1)) + 1/2)​
However, this contradicts your statement that f is a function from R to R, because the equation t = (t + 1/2)(t + 1/2) has no real solutions.

Also, the problem you stated contradicts what you said you got via the quadratic formula.

3. May 1, 2010