Question 163
Let $a$, $b$, $c$ be positive numbers satisfying:
$$
\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \le 16(a + b + c).
$$
Show that
$$
\frac{1}{(a + b + \sqrt{2(a + c)})^3} + \frac{1}{(b + c + \sqrt{2(b + a)})^3} + \frac{1}{(c + a + \sqrt{2(c + b)})^3} \le \frac{8}{9}.
$$
When does the equality occur?
$$
\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \le 16(a + b + c).
$$
Show that
$$
\frac{1}{(a + b + \sqrt{2(a + c)})^3} + \frac{1}{(b + c + \sqrt{2(b + a)})^3} + \frac{1}{(c + a + \sqrt{2(c + b)})^3} \le \frac{8}{9}.
$$
When does the equality occur?