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November 10th, 2015, 10:07 PM   #1
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Another inequality

I've been trying out more inequalities and this one tripped me up for a while. It's in the section on Cauchy-Schwarz and AM-GM 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.
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November 10th, 2015, 11:26 PM   #2
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Schwarz inequality and triangle inequality may be used to reach the destination
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November 11th, 2015, 12:22 AM   #3
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Have you solved it using those inequalities? If so, could you give me some pointers in that direction?
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