IMC 2012 · Problem 4

Let $n \ge 2$ be an integer. Find all real numbers $a$ such that there exist real numbers $x_1, \dots, x_n$ satisfying

$$x_1(1 - x_2) = x_2(1 - x_3) = \dots = x_{n-1}(1 - x_n) = x_n(1 - x_1) = a. \tag{1}$$