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Homework Help: Internat. Math Olympiad exercise. Weird function.

  1. May 1, 2010 #1
    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. jcsd
  3. May 1, 2010 #2


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    Staff Emeritus
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    Gold Member

    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.
  4. May 1, 2010 #3
    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) ) ).
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