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Real polynomial and system of equations

Source: Romanian IMO Team Selection Test TST 1987, problem 11

September 25, 2005
algebrapolynomialquadraticssystem of equationsalgebra proposed

Problem Statement

Let P(X,Y)=X2+2aXY+Y2P(X,Y)=X^2+2aXY+Y^2 be a real polynomial where ∣a∣≥1|a|\geq 1. For a given positive integer nn, n≥2n\geq 2 consider the system of equations: P(x1,x2)=P(x2,x3)=…=P(xn−1,xn)=P(xn,x1)=0. 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 (x1,x2,…,xn)(x_1,x_2,\ldots,x_n) and (y1,y2,…,yn)(y_1,y_2,\ldots,y_n) of the system to be equivalent if there exists a real number λ≠0\lambda \neq 0, x1=λy1x_1=\lambda y_1, …\ldots, xn=λynx_n= \lambda y_n. How many nonequivalent solutions does the system have? Mircea Becheanu
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