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Prove the second inequality assuming the first

  1. Nov 8, 2007 #1
    1. The problem statement, all variables and given/known data

    http://www.math.cornell.edu/~putnam/ineqs.pdf

    Can someone give me a hint on problem 1?

    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 9, 2007 #2

    Hurkyl

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    You were already given a hint: prove the second inequality assuming the first. Second hint below:




















    Isn't the expression

    [tex]\left( \frac{ \frac{1}{a_1} + \cdots + \frac{1}{a_n} }{n} \right)^{-1}[/tex]

    already in a form to which the first inequality is applicable?
     
    Last edited: Nov 9, 2007
  4. Nov 9, 2007 #3
    Yes. I read the problem. I spent 30 minutes manipulating both the first and the second inequality. Maybe I am missing something obvious, but I just don't see how to manipulate correctly. I tried logarithms, finding common denominators...
     
  5. Nov 9, 2007 #4

    Hurkyl

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    Bah, you saw it before I edited my post. :tongue:
     
  6. Nov 9, 2007 #5
    Wow. I feel stupid now. :)
     
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