Can you solve this inequality without using AM-GM?

[lakshya91]

I have no idea how to solve this inequality without using AM-GM inequality:


for any positive integers p,q,x,y show that
(pq+xy)(px+qy)>=4pqxy

 

[mrbohn1]

Multiply out the brackets on the left hand side, and then divide everything by pqxy.

You are left with: p/y + y/p + q/x + x/q on the left. All there is left to do to prove the result is to show that for any positive integers x and y: x/y + y/x >= 2.

 

[mathman]

In general for x > 0, x + 1/x ≥ 2.
This is trivially proven by x² -2x + 1 = (x-1)² ≥ 0.