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Thread: TA’s Challenge Problem #4

  1. June 19th 2009, 06:12 AM #1
    TheAbstractionist
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    TA’s Challenge Problem #4

    Let a_1,a_2,\ldots,a_n,r,s be positive integers such that rs\ge\frac1{n^4}. Show that

    \sum_{1\,\le\,i,j\,\le\,n}\left(\frac{ra_i}{a_j}+\ frac{sa_j}{a_i}\right)\ \geqslant\ 2
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  2. June 19th 2009, 06:24 AM #2
    malaygoel
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    Quote Originally Posted by TheAbstractionist View Post
    \sum_{1\,\le\,i,j\,\le\,n}\left(\frac{ra_i}{a_j}+\ frac{sa_j}{a_i}\right)\ \geqslant\ 2
    There will be 2n^2 terms in the summation.

    Apply AM-GM inequality
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  3. June 21st 2009, 02:08 AM #3
    TheAbstractionist
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    Yes, AM–GM will work.

    When I made the problem, I didn’t think of AM–GM and so I thought the problem was more challenging than it’s actually turned out to be.
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