Question 13
Determine the smallest and the greatest possible values of the expression
$$
\left(\frac{1}{a^{2}+1}+\frac{1}{b^{2}+1}+\frac{1}{c^{2}+1}\right)\left(\frac{a^{2}}{a^{2}+1}+\frac{b^{2}}{b^{2}+1}+\frac{c^{2}}{c^{2}+1}\right)
$$
provided $a$, $b$, and $c$ are non-negative real numbers satisfying $ab+bc+ca=1$.
$$
\left(\frac{1}{a^{2}+1}+\frac{1}{b^{2}+1}+\frac{1}{c^{2}+1}\right)\left(\frac{a^{2}}{a^{2}+1}+\frac{b^{2}}{b^{2}+1}+\frac{c^{2}}{c^{2}+1}\right)
$$
provided $a$, $b$, and $c$ are non-negative real numbers satisfying $ab+bc+ca=1$.