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Math Help - Problem 34

  1. #1
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    Problem 34

    Given x,y,z\in \mathbb{Z} solve the Diophantine equation:
    x^3+y^3+z^3 = (x+y+z)^3.
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  2. #2
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    All permutations of (k, -k, j) for integers k and j are solutions. To see that these are the only solutions, rewrite as y^3+z^3 = (x+y+z)^3 - x^3 which, assuming  y \ne -z, reduces to x^2+(y+z)x+yz = 0 \Rightarrow x = -y or x = -z with the other variable arbitrary, giving the solutions (k, -k, j), \ (k, j, -k) and, by symmetry in the original equation, (j, k, -k)
    Last edited by mathisfun1; August 14th 2007 at 03:11 PM.
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  3. #3
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    Is it possible to propose the next 'problem of the week'?
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  4. #4
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    Apparently I can't post new threads, so I'll just append a problem here.

    Source: The Art of Problem Solving (Vol 2)

    Show that for any two positive numbers, RMS-AM >= GM-HM where RMS denotes the root-mean square, AM the arithmetic mean, GM the geometric mean, and HM the harmonic mean.
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  5. #5
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    Quote Originally Posted by albi View Post
    Is it possible to propose the next 'problem of the week'?
    I try to get one by Sunday or Monday.

    Apparently I can't post new threads, so I'll just append a problem here.
    Blocked priveleges.
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  6. #6
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    Quote Originally Posted by ThePerfectHacker View Post
    I try to get one by Sunday or Monday.
    Um. I think we didnt understand ourselves.

    I meant I have one idea which may become ´problem of the week´.
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  7. #7
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    Quote Originally Posted by albi View Post
    Um. I think we didnt understand ourselves.

    I meant I have one idea which may become ´problem of the week´.
    Of course you can! We're always open to suggestions. The Problem of the Week writers (CB and TPH) work very hard though so be constructive!
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