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System of equations with sequence - ISL 1986

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August 31, 2010
algebrapolynomialsystem of equationscalculusIMO Shortlist

Problem Statement

Let real numbers x1,x2,⋯ ,xnx_1, x_2, \cdots , x_n satisfy 0<x1<x2<⋯<xn<10 < x_1 < x_2 < \cdots< x_n < 1 and set x0=0,xn+1=1x_0 = 0, x_{n+1} = 1. Suppose that these numbers satisfy the following system of equations: \sum_{j=0, j \neq i}^{n+1} \frac{1}{x_i-x_j}=0   \text{where } i = 1, 2, . . ., n. Prove that xn+1−i=1−xix_{n+1-i} = 1- x_i for i=1,2,...,n.i = 1, 2, . . . , n.
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