Question 92
Let $a, b, c \gt 0$ and $a + b + c = 3$. Prove that
$$
\frac{1}{a\sqrt{2(a^2+bc)}} + \frac{1}{b\sqrt{2(b^2+ca)}} + \frac{1}{c\sqrt{2(c^2+ab)}} \ge \frac{1}{a+bc} + \frac{1}{b+ca} + \frac{1}{c+ab}.
$$
$$
\frac{1}{a\sqrt{2(a^2+bc)}} + \frac{1}{b\sqrt{2(b^2+ca)}} + \frac{1}{c\sqrt{2(c^2+ab)}} \ge \frac{1}{a+bc} + \frac{1}{b+ca} + \frac{1}{c+ab}.
$$