<s>[INST] How do I show that $\frac {a^2}b + \frac {b^2}c + \frac {c^2}a \ge \frac {(a + b + c)(a^2 + b^2 + c^2)}{ab + bc + ca}?$ For positive real numbers $a, b, c$, show that
$$\frac {a^2}b + \frac {b^2}c + \frac {c^2}a \ge \frac {(a + b + c)(a^2 + b^2 + c^2)}{ab + bc + ca}.$$
I don't know how to solve this at all. Can you provide any hints?
 [/INST] We have to prove that $\implies$
$\sum_{cyclic}\frac{a^2}{b}\geq{\frac{(a+b+c)(a^2+b^2+c^2)}{ab+bc+ca}}$
$\implies (ab+bc+ca)(\sum_{cyclic}\frac{a^2}{b})\geq(a+b+c)(a^2+b^2+c^2)$
Now expand both sides and many terms are cancelled out.Then we get the 
inequality $\implies(\sum_{cyclic}\frac{ca^3}{b})\geq(ac^2+ba^2+cb^2)$
Now use the AM-GM inequality to
1.$(\frac{ca^3}{b}\;,\frac{ab^3}{c})$
2.$(\frac{ab^3}{c}\;,\frac{bc^3}{a})$
3.$(\frac{bc^3}{a}\;,\frac{ca^3}{b})$
Now the required inequality is proved.
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