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Prove : (a^2+b^2+c^2+d^2)(1/a^2+1/b^2+1/c^2+1/d^2)greater equal to 16

  • maths -

    a^2 * 1/a^2 >= 1

    assume
    (a1^2+...+ak^2)(1/a1^2+...+1/ak^2) >= k^2

    (a1^2+...+ak+1^2)(1/a1^2+...+1/ak+1^2)
    = (a1^2+...+ak^2)(1/a1^2+...+1/ak+1^2)
    + ak+1^2((1/a1^2+...+1/ak+1^2)
    = (a1^2+...+ak^2)(1/a1^2+...+1/ak^2)
    + (a1^2+...+ak^2)*1/ak+1^2
    + ak+1^2((1/a1^2+...+1/ak^2)
    + ak+1^2 * 1/ak+1^2
    now, by hypothesis,
    >= k^2
    + (a1^2+...+ak^2)*1/ak+1^2
    + ak+1^2((1/a1^2+...+1/ak^2)
    + 1
    Hmmm. We have to prove that the two middle terms >= 2k, then we have proved the induction step. Gotta go now, but this may help some.

    If you can show it, then the given problem is just a special case where k=4.

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