MathDB
2017 Shortlist/A5

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July 10, 2018
algebrainequalitiesIMO Shortlist

Problem Statement

An integer n≥3n \geq 3 is given. We call an nn-tuple of real numbers (x1,x2,…,xn)(x_1, x_2, \dots, x_n) Shiny if for each permutation y1,y2,…,yny_1, y_2, \dots, y_n of these numbers, we have ∑i=1n−1yiyi+1=y1y2+y2y3+y3y4+⋯+yn−1yn≥−1.\sum \limits_{i=1}^{n-1} y_i y_{i+1} = y_1y_2 + y_2y_3 + y_3y_4 + \cdots + y_{n-1}y_n \geq -1. Find the largest constant K=K(n)K = K(n) such that ∑1≤i<j≤nxixj≥K\sum \limits_{1 \leq i < j \leq n} x_i x_j \geq K holds for every Shiny nn-tuple (x1,x2,…,xn)(x_1, x_2, \dots, x_n).
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