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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Originally Posted by TheAbstractionist $\displaystyle \sum_{1\,\le\,i,j\,\le\,n}\left(\frac{ra_i}{a_j}+\ frac{sa_j}{a_i}\right)\ \geqslant\ 2$ There will be $\displaystyle 2n^2$ terms in the summation. Apply AM-GM inequality
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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