# TA’s Challenge Problem #4

• Jun 19th 2009, 07:12 AM
TheAbstractionist
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

$\sum_{1\,\le\,i,j\,\le\,n}\left(\frac{ra_i}{a_j}+\ frac{sa_j}{a_i}\right)\ \geqslant\ 2$
• Jun 19th 2009, 07:24 AM
malaygoel
Quote:

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

There will be $2n^2$ terms in the summation.

Apply AM-GM inequality
• Jun 21st 2009, 03:08 AM
TheAbstractionist
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)