MathDB
Cyclic system of equations with many solutions

Source: VJIMC 2017, Category II, Problem 3

April 2, 2017
algebrasystem of equations

Problem Statement

Let n≥2n \ge 2 be an integer. Consider the system of equations \begin{align} x_1+\frac{2}{x_2}=x_2+\frac{2}{x_3}=\dots=x_n+\frac{2}{x_1} \end{align} 1. Prove that (1)(1) has infinitely many real solutions (x1,…,xn)(x_1,\dotsc,x_n) such that the numbers x1,…,xnx_1,\dotsc,x_n are distinct. 2. Prove that every solution of (1)(1), such that the numbers x1,…,xnx_1,\dotsc,x_n are not all equal, satisfies ∣x1x2⋯xn∣=2n/2\vert x_1x_2\cdots x_n\vert=2^{n/2}.
Back to ProblemsView on AoPS