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x_1 + x_2 + ... + x_n >= 0

Source: IMO Shortlist 2010, Algebra 4

July 17, 2011
algebraInequalitySequenceIMO Shortlistget the smallest

Problem Statement

A sequence x1,x2,…x_1, x_2, \ldots is defined by x1=1x_1 = 1 and x2k=−xk,x2k−1=(−1)k+1xkx_{2k}=-x_k, x_{2k-1} = (-1)^{k+1}x_k for all k≥1.k \geq 1. Prove that ∀n≥1\forall n \geq 1 x1+x2+…+xn≥0.x_1 + x_2 + \ldots + x_n \geq 0.
Proposed by Gerhard Wöginger, Austria
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