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November 10th, 2015, 10:07 PM  #1 
Math Team Joined: Nov 2014 From: Australia Posts: 689 Thanks: 244  Another inequality
I've been trying out more inequalities and this one tripped me up for a while. It's in the section on CauchySchwarz and AMGM has been covered beforehand. The problem is: For $a,b,c,d > 0$ and $a^2 + b^2 + c^2 + d^2 = 4$, prove that $$ \dfrac{a^2}{b} + \dfrac{b^2}{c} + \dfrac{c^2}{d} + \dfrac{d^2}{a}\geq 4 $$ I have a solution, but it's very messy. I'm wondering whether there is a nice way to solve this using only the two inequalities mentioned above. 
November 10th, 2015, 11:26 PM  #2 
Newbie Joined: Aug 2015 From: USA Posts: 29 Thanks: 0 
Schwarz inequality and triangle inequality may be used to reach the destination

November 11th, 2015, 12:22 AM  #3 
Math Team Joined: Nov 2014 From: Australia Posts: 689 Thanks: 244 
Have you solved it using those inequalities? If so, could you give me some pointers in that direction?


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