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Math Help - Arithmetic mean and Geometric mean

  1. #1
    MHF Contributor alexmahone's Avatar
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    Arithmetic mean and Geometric mean

    The difference between the arithmetic mean and the geometric mean of two positive integers m and n is 1. Prove that \frac {m}{2} and \frac {n}{2} are perfect squares.
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  2. #2
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    \frac {m + n}{2} = 1 + \sqrt {mn}

    Clearly mn is a perfect square, therefore m = a^2 t, \, n = b^2 t for some square free t

    Therefore

    \frac {t(a^2 + b^2)}{2} = 1 + abt \Rightarrow t(a - b)^2 = 2 \Rightarrow t = 2 as required
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