<s>[INST] How do we prove this inequality? Suppose $a,b,c > 0$. Prove that
$$\frac{a^2}{b^2} +\frac{b^2}{c^2} + \frac{c^2}{a^2} \geq \frac ab + \frac bc + \frac ca.$$
I've tried multiplying everything by the denominator and then I tried to use the rearrangement inequality, but it didn't yield the result I was looking for. I couldn't really think of anything else to do. 
 [/INST] After clearing the denominators we have to prove that
$$a^4c^2+b^4a^2+b^2c^4\geq abc(a^2c+ab^2+bc^2)$$
Now use that $$x^2+y^2+z^2\geq xy+yz+zx$$
This is $$(a^2c)^2+(bc^2)^2+(ab^2)^2\geq a^2cab^2+a^2cbc^2+ab^2bc^2=abc(a^2b+ac^2+b^2c)$$
Since$$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=\frac{a^2c+ab^2+a^2b}{abc}$$
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