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

$\displaystyle \sum_{1\,\le\,i,j\,\le\,n}\left(\frac{ra_i}{a_j}+\ frac{sa_j}{a_i}\right)\ \geqslant\ 2$

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- Jun 19th 2009, 06:12 AMTheAbstractionistTA’s Challenge Problem #4
Let $\displaystyle a_1,a_2,\ldots,a_n,r,s$ be positive integers such that $\displaystyle rs\ge\frac1{n^4}.$ Show that

$\displaystyle \sum_{1\,\le\,i,j\,\le\,n}\left(\frac{ra_i}{a_j}+\ frac{sa_j}{a_i}\right)\ \geqslant\ 2$ - Jun 19th 2009, 06:24 AMmalaygoel
- Jun 21st 2009, 02:08 AMTheAbstractionist
Yes, AM–GM will work. (Clapping)

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. (Giggle)