MathNet 2004 — Part 01 Question 28

Let $a, b, c \gt 0$ and $a + b + c = 1$. Prove that
$$
\frac{9}{10} \leq \frac{a}{1 + b c} + \frac{b}{1 + c a} + \frac{c}{1 + a b} \lt 1
$$