Question 123
(A) Prove that for all real numbers $\kappa, \lambda, \mu$ holds the following inequality:
$$ (\kappa + \lambda + \mu)^2 \ge 3(\kappa\lambda + \lambda\mu + \mu\kappa). $$
When is equality valid?
(B) Let $x, y, z$ be positive real numbers and let $\alpha, \beta$ be real numbers such that
$$
\alpha(x + y + z) = \beta(xy + yz + zx) = xyz.
$$
Prove that $\alpha \ge 3\beta^2$. When is equality valid?
$$ (\kappa + \lambda + \mu)^2 \ge 3(\kappa\lambda + \lambda\mu + \mu\kappa). $$
When is equality valid?
(B) Let $x, y, z$ be positive real numbers and let $\alpha, \beta$ be real numbers such that
$$
\alpha(x + y + z) = \beta(xy + yz + zx) = xyz.
$$
Prove that $\alpha \ge 3\beta^2$. When is equality valid?