Question 53
Show that
$$
\frac{1}{ab + 2c^2 + 2c} + \frac{1}{bc + 2a^2 + 2a} + \frac{1}{ca + 2b^2 + 2b} \geq \frac{1}{ab + bc + ca}
$$
for all positive real numbers $a$, $b$, $c$ satisfying the equality $a + b + c = 1$.
$$
\frac{1}{ab + 2c^2 + 2c} + \frac{1}{bc + 2a^2 + 2a} + \frac{1}{ca + 2b^2 + 2b} \geq \frac{1}{ab + bc + ca}
$$
for all positive real numbers $a$, $b$, $c$ satisfying the equality $a + b + c = 1$.