Question 53
Let $a$, $b$, $c$ be positive real numbers such that $a^{2} + b^{2} + c^{2} = 1$. Find the minimum for
$$
S = \frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}} - \frac{2\left(a^{3} + b^{3} + c^{3}\right)}{a b c}
$$
$$
S = \frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}} - \frac{2\left(a^{3} + b^{3} + c^{3}\right)}{a b c}
$$