prove that $\frac{b+c}{a^2}+\frac{c+a}{b^2}+\frac{a+b}{c^2}\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ Assume: $a,b,c >0$ prove that :
$$\frac{b+c}{a^2}+\frac{c+a}{b^2}+\frac{a+b}{c^2}\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$$

HINT: One may start proving the inequality:
$$\frac{b+c}{a^2}+\frac{c+a}{b^2}+\frac{a+b}{c^2}\ge\ \frac{9}{a+b+c}+ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}$$
that is easy to prove. I met some time ago this inequality and have just remembered now. Of course, it's easy only if you met it before, otherwise it's rather hard to make such a guess.
