$$\sum_{cyc} \sqrt{(a^2+ab+bc+ca)(b^2+ab+bc+ca)} \geq \sum_{cyc} (ab+ab+bc+ca)$$
$$2\sum_{cyc} \frac{a^3b}{a^4+a^2b^2+b^4}\leq 2\sum_{cyc} \frac{a^3b}{2a^3b+b^4} =3- \sum_{cyc} \frac{b^3}{2a^3+b^3} \leq 3-1=2$$
$$\sum_{cyc} \frac{a^3}{b+c}>\sum_{cyc} \frac{a}{b+c}\geq\frac{3}{2}$$
$$\underbrace{\frac{1}{2^{n+1}}+\frac{1}{2^{n+1}}+...+\frac{1}{2^{n+1}}}_{2^n ตัว} =\frac{1}{2}$$