AUSTRIAN-POLISH 2000 Problem 9

If three nonnegative reals $a$ , $b$ , $c$ satisfy $a+b+c=1$ , prove that

$2 \leq \left(1-a^{2}\right)^{2}+\left(1-b^{2}\right)^{2}+\left(1-c^{2}\right)^{2}\leq \left(1+a\right)\left(1+b\right)\left(1+c\right)$ .