Question 68
Let $a, b, c \gt 0$ and $abc = 1$. Prove that
$$
\frac{a^{2014}}{1+2bc} + \frac{b^{2014}}{1+2ca} + \frac{c^{2014}}{1+2ab} \ge \frac{3}{ab+bc+ca}
$$
$$
\frac{a^{2014}}{1+2bc} + \frac{b^{2014}}{1+2ca} + \frac{c^{2014}}{1+2ab} \ge \frac{3}{ab+bc+ca}
$$