Question 28
Determine the largest constant $C$ such that
$$
(x_1 + x_2 + \dots + x_6)^2 \geq C \cdot (x_1(x_2 + x_3) + x_2(x_3 + x_4) + \dots + x_6(x_1 + x_2))
$$
holds for all real numbers $x_1, x_2, \dots, x_6$.
For this $C$, determine all $x_1, x_2, \dots, x_6$ such that equality holds.
$$
(x_1 + x_2 + \dots + x_6)^2 \geq C \cdot (x_1(x_2 + x_3) + x_2(x_3 + x_4) + \dots + x_6(x_1 + x_2))
$$
holds for all real numbers $x_1, x_2, \dots, x_6$.
For this $C$, determine all $x_1, x_2, \dots, x_6$ such that equality holds.