Question 172
Let $a$, $b$, $c$ be positive real numbers. Prove that
$$
\frac{1}{a(1+b)} + \frac{1}{b(1+c)} + \frac{1}{c(1+a)} \geq \frac{3}{1+abc},
$$
and that equality occurs if and only if $a = b = c = 1$.
$$
\frac{1}{a(1+b)} + \frac{1}{b(1+c)} + \frac{1}{c(1+a)} \geq \frac{3}{1+abc},
$$
and that equality occurs if and only if $a = b = c = 1$.