Question 4
Find the largest constant $C$ such that
$$\sum_{i=1}^4\left(x_i+\frac1{x_i}\right)^3\ge C$$
for all positive real numbers $x_1,\ldots,x_4$ satisfying
$$x_1^3+x_3^3+3x_1x_3=x_2+x_4=1.$$
$$\sum_{i=1}^4\left(x_i+\frac1{x_i}\right)^3\ge C$$
for all positive real numbers $x_1,\ldots,x_4$ satisfying
$$x_1^3+x_3^3+3x_1x_3=x_2+x_4=1.$$