MathDB

6

Part of 2001 South africa National Olympiad

Problems(1)

System with unique solution

Source: South Africa 2001

10/1/2005
The unknown real numbers x1,x2,…,xnx_1,x_2,\dots,x_n satisfy x1<x2<⋯<xn,x_1 < x_2 < \cdots < x_n, where n≥3n \geq 3. The numbers ss, tt and d1,d2,…,dn−2d_1,d_2,\dots,d_{n - 2} are given, such that s=∑i=1nxi,t=∑i=1nxi2,di=xi+2−xi,  i=1,2,…,n−2. \begin{aligned} s & = \sum\limits_{i = 1}^nx_i, \\ t & = \sum\limits_{i = 1}^nx_i^2,\\ d_i & = x_{i + 2} - x_i,\ \ i = 1,2,\dots,n - 2. \end{aligned} For which nn is this information always sufficient to determine x1,x2,…,xnx_1,x_2,\dots,x_n uniquely?
algebra unsolvedalgebra