IMC 2012 Problem 9

Let $n \ge 2$ be an integer. Find all real numbers $a$ such that there exist real numbers $x_1,x_2,\dots,x_n$ satisfying $$x_1(1-x_2)=x_2(1-x_3)=\dots=x_n(1-x_1)=a.$$

Proposed by Walther Janous and Gerhard Kirchner, Innsbruck.