AT-OEMO-FED1 2016 Problem 1

Determine the largest constant $C$ such that $$(x_1 + x_2 + \cdots + x_6)^2 \ge C \cdot (x_1(x_2 + x_3) + x_2(x_3 + x_4) + \cdots + x_6(x_1 + x_2))$$ holds for all real numbers $x_1, x_2, \cdots , x_6$ .

For this $C$ , determine all $x_1, x_2, \cdots x_6$ such that equality holds.

(Walther Janous)