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Thread: Inequality help

  1. #1
    Jul 2013
    New Delhi

    Inequality help

    Wasn't sure where to put this...profound apologies if this is in the wrong forum...

    Suppose there are $n$ distinct odd, positive integers $\{a_i\}_{i=1}^n$ where the absolute value of the difference between each pair of numbers is distinct i.e. each $|a_k - a_i|, \ 1\leq k < i \leq n$ is distinct.

    Prove that $$\sum_{i=1}^n a_i \geq \tfrac{1}{3} n(n^2 + 2)$$

    I tried doing this by induction and have gotten to the stage of my induction hypothesis but am not sure how to proceed. I have tried to prove that $a_{k+1} \geq 1 + k + k^2$ (as this would help yield the result for my induction) but, again, no success....

    Can anyone give me a hint as to how I would go about the induction (or even if induction is the correct way and I should try a different approach)?

    Thank you
    Last edited by Diadem; Feb 23rd 2014 at 03:07 AM.
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  2. #2
    MHF Contributor
    Dec 2012
    Athens, OH, USA

    Re: Inequality help

    You're definitely on the right track. The attachment shows an outline of a proof. If you have any questions, post again.

    Inequality help-mhfnumbertheory7.png
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