Question 54
Answer the maximum value of $A$ for which, for every positive $x_1, x_2, x_3, y_1, y_2, y_3, z_1, z_2$ and $z_3$, the inequality
$$
(x_1^3 + x_2^3 + x_3^3 + 1)(y_1^3 + y_2^3 + y_3^3 + 1)(z_1^3 + z_2^3 + z_3^3 + 1)
\geq A(x_1 + y_1 + z_1)(x_2 + y_2 + z_2)(x_3 + y_3 + z_3)
$$
holds.
For the maximum value of $A$, establish the cases of equality.
$$
(x_1^3 + x_2^3 + x_3^3 + 1)(y_1^3 + y_2^3 + y_3^3 + 1)(z_1^3 + z_2^3 + z_3^3 + 1)
\geq A(x_1 + y_1 + z_1)(x_2 + y_2 + z_2)(x_3 + y_3 + z_3)
$$
holds.
For the maximum value of $A$, establish the cases of equality.