RO-TST 1987 Problem 11

Let $P(X,Y)=X^2+2aXY+Y^2$ be a real polynomial where $|a|\geq 1$ . For a given positive integer $n$ , $n\geq 2$ consider the system of equations: $$P(x_1,x_2) = P(x_2,x_3) = \ldots = P(x_{n-1},x_n) = P(x_n,x_1) = 0 .$$ We call two solutions $(x_1,x_2,\ldots,x_n)$ and $(y_1,y_2,\ldots,y_n)$ of the system to be equivalent if there exists a real number $\lambda \neq 0$ , $x_1=\lambda y_1$ , $\ldots$ , $x_n= \lambda y_n$ . How many nonequivalent solutions does the system have?

Mircea Becheanu