Internat. Math Olympiad exercise. Weird function.

[GBarboza]

Homework Statement

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.

Homework 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.

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.

 

[Hurkyl]

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.

 

[GBarboza]

Thanks for replying.

Right, I'm pretty sure that the function being from R to R is the part that I got wrong. He told me that later in the day(or, at least, that's what I managed to make of the three or so words he said); earlier he had told me to find the possible values of x, which is what I can't do. :/

Oh, and, oops, I meant f(x) = -1/2 +- sq( f( x + xy + f(y) ) ).