PL-OM 2011 Problem 3

Let $n\geq 3$ be an odd integer. Determine how many real solutions there are to the set of $n$ equations $$\left\{\begin{array}{cc}x_1(x_1+1)=x_2(x_2-1)x_2(x_2+1)=x_3(x_3-1) \vdots x_n(x_n+1) = x_1(x_1-1)\end{array}\right.$$