Question 146
Prove that, if $a, b, c \geq 0$ and $a+b+c=3$, then
$$
\frac{a}{1+b}+\frac{b}{1+c}+\frac{c}{1+a} \geq \frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}
$$
$$
\frac{a}{1+b}+\frac{b}{1+c}+\frac{c}{1+a} \geq \frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}
$$