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Turkey NMO 2017 p5

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January 25, 2018
algebrainequalities

Problem Statement

Let x0,…,x2017x_0,\dots,x_{2017} are positive integers and x2017≥⋯≥x0=1x_{2017}\geq\dots\geq x_0=1 such that A={x1,…,x2017}A=\{x_1,\dots,x_{2017}\} consists of exactly 2525 different numbers. Prove that ∑i=22017(xi−xi−2)xi≥623\sum_{i=2}^{2017}(x_i-x_{i-2})x_i\geq 623, and find the number of sequences that holds the case of equality.
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