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Math Help - Set of 13 real numbers

  1. #1
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    Set of 13 real numbers

    Hi, I'm having difficulty getting started with this problem:

    Show that given any set of 13 distinct real numbers, there will be two elements in the set (x and y) such that
    0 < \frac{x-y}{1+xy} \leq 2 - \sqrt{3}

    I really don't know where to start, but this is a course in combinatorics, and the pigeonhole principle is supposed to be used somewhere (probably in the part about sets)

    1) How does the set having 13 elements come into play? I know that if we're talking about integers, then it would probably be related to some modulo thing (forcing 2 elements to have the same residue) but we're talking about real numbers

    2) What is so special about the expression \frac{x-y}{1+xy}? I mean, is this an expression of a property? Like distance formula or something? The closest thing I can get to is the difference of inverses (i.e. \frac{1}{y} - \frac{1}{x} = \frac{x-y}{xy}, but I don't know where the "1+" part is coming from.

    3) What's the significance of 2 - \sqrt{3}? Could it be replaced by anything else?

    Basically, I get what's important, but I don't get what's connecting them ... any help with those connections would be great

    Thanks in advance!
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Set of 13 real numbers

    Quote Originally Posted by Bingk View Post
    Hi, I'm having difficulty getting started with this problem:

    Show that given any set of 13 distinct real numbers, there will be two elements in the set (x and y) such that
    0 < \frac{x-y}{1+xy} \leq 2 - \sqrt{3}

    I really don't know where to start, but this is a course in combinatorics, and the pigeonhole principle is supposed to be used somewhere (probably in the part about sets)

    1) How does the set having 13 elements come into play? I know that if we're talking about integers, then it would probably be related to some modulo thing (forcing 2 elements to have the same residue) but we're talking about real numbers

    2) What is so special about the expression \frac{x-y}{1+xy}? I mean, is this an expression of a property? Like distance formula or something? The closest thing I can get to is the difference of inverses (i.e. \frac{1}{y} - \frac{1}{x} = \frac{x-y}{xy}, but I don't know where the "1+" part is coming from.

    3) What's the significance of 2 - \sqrt{3}? Could it be replaced by anything else?

    Basically, I get what's important, but I don't get what's connecting them ... any help with those connections would be great

    Thanks in advance!
    Hint(I think):




    Define x_i:=tan(\alpha_i) , \alpha_i \in (-\frac{\pi}{2},\frac{\pi}{2}) for all 1\leq i \leq 13 and use Dirichlet's principle.
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  3. #3
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    Re: Set of 13 real numbers

    Thanks! I didn't think about using the tangent function, but everything came together with that
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