MathNet 2020 — Math Olympiad Vault — Part 05 Question 106

If $a^{3}+b^{3}+c^{3}=3 a b c=6$ and $a^{2}+b^{2}+c^{2}=8$, find the value of
$$
\frac{a b}{a+b}+\frac{b c}{b+c}+\frac{c a}{c+a}
$$