Question 88
Show that for all positive real numbers $a$, $b$, $c$ satisfying $a + b + c = 1$ the following inequality is held:
$$
\frac{a^4 + 5b^4}{a(a + 2b)} + \frac{b^4 + 5c^4}{b(b + 2c)} + \frac{c^4 + 5a^4}{c(c + 2a)} \ge 1 - ab - bc - ca
$$
$$
\frac{a^4 + 5b^4}{a(a + 2b)} + \frac{b^4 + 5c^4}{b(b + 2c)} + \frac{c^4 + 5a^4}{c(c + 2a)} \ge 1 - ab - bc - ca
$$