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China TST 2010, Problem 6

Source:

August 28, 2010
inequalitiesinductionalgebra unsolvedalgebra

Problem Statement

Given integer n≥2n\geq 2 and real numbers x1,x2,⋯ ,xnx_1,x_2,\cdots, x_n in the interval [0,1][0,1]. Prove that there exist real numbers a0,a1,⋯ ,ana_0,a_1,\cdots,a_n satisfying the following conditions: (1) a0+an=0a_0+a_n=0; (2) ∣ai∣≤1|a_i|\leq 1, for i=0,1,⋯ ,ni=0,1,\cdots,n; (3) ∣ai−ai−1∣=xi|a_i-a_{i-1}|=x_i, for i=1,2,⋯ ,ni=1,2,\cdots,n.
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