olymmath

ISL 2010 A4

    A sequence x1,x2,…x_{1}, x_{2}, \ldots is defined by x1=1x_{1}=1 and x2k=−xk,x2k−1=(−1)k+1xkx_{2 k}=-x_{k}, x_{2 k-1}=(-1)^{k+1} x_{k} for all k≥1k \geq 1. Prove that x1+x2+⋯+xn≥0x_{1}+x_{2}+\cdots+x_{n} \geq 0 for all n≥1n \geq 1. (Austria)

    Official text Transcription from NuminaMath olympiads, checked word for word against the official paper. Quoted with source.

    Official text: IMO 2010 Shortlist booklet, page 13.

    Solutions

    Official solution Solution 1 by the IMO Problem Selection Committee

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