Question 30
Suppose $x$, $y$ and $z$ are positive numbers such that
$$
1 = 2xyz + xy + yz + zx. \tag{1}
$$
Prove that
$$
(i) \quad \frac{3}{4} \le xy + yz + zx \lt 1;
$$
$$
(ii) \quad xyz \le \frac{1}{8}.
$$
Using (i) or otherwise, deduce that
$$
x + y + z \ge \frac{3}{2}, \tag{2}
$$
and derive the case of equality in (2).
$$
1 = 2xyz + xy + yz + zx. \tag{1}
$$
Prove that
$$
(i) \quad \frac{3}{4} \le xy + yz + zx \lt 1;
$$
$$
(ii) \quad xyz \le \frac{1}{8}.
$$
Using (i) or otherwise, deduce that
$$
x + y + z \ge \frac{3}{2}, \tag{2}
$$
and derive the case of equality in (2).