For all nonnegative real numbers $a,b$ and $c.$ Prove that:
$\color{blue}{\sum_{cyc}a^2\sum_{cyc}a(b+c)\sum_{cyc}\frac{1}{(b+c)^2}\geq (a+b+c)^4}$
Solution:
$\sum_{cyc}\frac{1}{(a+b)^2}\geq \frac{3(a+b+c)^2}{8(ab+bc+ca)}(\frac{1}{ab+bc+ca}+\frac{1}{a^2+b^2+c^2})$