MathDB

Suppose that

x, y

and

z

are positive real numbers such that

xyz \ge 1

. (a) Prove that 27 \le (1 \plus{} x \plus{} y)^2 \plus{} (1\plus{} x \plus{} z)^2 \plus{} (1 \plus{} y \plus{} z)^2, with equality if and only if x \equal{} y \equal{} z \equal{} 1. (b) Prove that (1 \plus{} x \plus{} y)^2 \plus{} (1\plus{} x \plus{} z)^2 \plus{} (1 \plus{} y \plus{} z)^2 \le 3(x \plus{} y \plus{} z)^2, with equality if and only if x \equal{} y \equal{} z \equal{} 1.

Problem Statement