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Cyclic system of equations

Source: Bulgaria NMO 2022 P4

April 17, 2022
algebrasystem of equationsparameterBulgaria

Problem Statement

Let n≥4n\geq 4 be a positive integer and x1,x2,…,xn,xn+1,xn+2x_{1},x_{2},\ldots ,x_{n},x_{n+1},x_{n+2} be real numbers such that xn+1=x1x_{n+1}=x_{1} and xn+2=x2x_{n+2}=x_{2}. If there exists an a>0a>0 such that x_{i}^2=a+x_{i+1}x_{i+2} \forall 1\leq i\leq n then prove that at least 22 of the numbers x1,x2,…,xnx_{1},x_{2},\ldots ,x_{n} are negative.
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