MEMO 2019 T1

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 $ab+bc+ca=1$ .

Proposed by Walther Janous, Austria