Question 81
Let $a$, $b$, $c$ be real numbers and let
$$
x = a + b + c, \quad y = a^2 + b^2 + c^2, \quad z = a^3 + b^3 + c^3 \quad \text{and} \quad S = 2x^3 - 9xy + 9z.
$$
a. Prove that $S$ is unchanged when $a$, $b$, $c$ are replaced by $a + t$, $b + t$, $c + t$, respectively, for any real number $t$.
b. Prove that $(3y - x^2)^3 \ge 2S^2$.
$$
x = a + b + c, \quad y = a^2 + b^2 + c^2, \quad z = a^3 + b^3 + c^3 \quad \text{and} \quad S = 2x^3 - 9xy + 9z.
$$
a. Prove that $S$ is unchanged when $a$, $b$, $c$ are replaced by $a + t$, $b + t$, $c + t$, respectively, for any real number $t$.
b. Prove that $(3y - x^2)^3 \ge 2S^2$.