Question 13
Prove that the inequality
$$
\left(\frac{a^2 + b^2}{a + b}\right)^3 + \left(\frac{b^2 + c^2}{b + c}\right)^3 + \left(\frac{c^2 + a^2}{c + a}\right)^3 \ge a^3 + b^3 + c^3
$$
holds for all $a, b, c \gt 0$.
$$
\left(\frac{a^2 + b^2}{a + b}\right)^3 + \left(\frac{b^2 + c^2}{b + c}\right)^3 + \left(\frac{c^2 + a^2}{c + a}\right)^3 \ge a^3 + b^3 + c^3
$$
holds for all $a, b, c \gt 0$.