Question 180
Suppose that $x$, $y$ and $z$ are positive real numbers such that $xyz \ge 1$.
a. Prove that
$$
27 \le (1+x+y)^2 + (1+y+z)^2 + (1+z+x)^2,
$$
with equality if and only if $x = y = z = 1$.
b. Prove that
$$
(1+x+y)^2 + (1+y+z)^2 + (1+z+x)^2 \le 3(x+y+z)^2,
$$
with equality if and only if $x = y = z = 1$.
a. Prove that
$$
27 \le (1+x+y)^2 + (1+y+z)^2 + (1+z+x)^2,
$$
with equality if and only if $x = y = z = 1$.
b. Prove that
$$
(1+x+y)^2 + (1+y+z)^2 + (1+z+x)^2 \le 3(x+y+z)^2,
$$
with equality if and only if $x = y = z = 1$.