Question 18
Show that for all positive real numbers $a, b, c$ satisfying $a^3 + b^3 + c^3 = a^4 + b^4 + c^4$ the following inequality is held:
$$
\frac{a}{a^2 + b^3 + c^3} + \frac{b}{a^3 + b^2 + c^3} + \frac{c}{a^3 + b^3 + c^2} \ge 1
$$
$$
\frac{a}{a^2 + b^3 + c^3} + \frac{b}{a^3 + b^2 + c^3} + \frac{c}{a^3 + b^3 + c^2} \ge 1
$$