Question 130
The positive real numbers $a$, $b$, $c$ satisfy: $a^2 + b^2 + c^2 = 3$. Prove that
$$
\frac{a^2 + b^2}{2ab} + \frac{b^2 + c^2}{2bc} + \frac{c^2 + a^2}{2ca} + \frac{2(ab + bc + ca)}{3} \geq 5.
$$
When does equality hold?
$$
\frac{a^2 + b^2}{2ab} + \frac{b^2 + c^2}{2bc} + \frac{c^2 + a^2}{2ca} + \frac{2(ab + bc + ca)}{3} \geq 5.
$$
When does equality hold?