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inequality, sum of (x_n^2)/(n^3)

Source: Serbia & Montenegro 2001 3,4th Grade P2

June 2, 2021
inequalitiesSummation

Problem Statement

Let x1,x2,…,x2001x_1,x_2,\ldots,x_{2001} be positive numbers such that xi2≥x12+x2223+x3233+…+xi−12(i−1)3for 2≤i≤2001.x_i^2\ge x_1^2+\frac{x_2^2}{2^3}+\frac{x_3^2}{3^3}+\ldots+\frac{x_{i-1}^2}{(i-1)^3}\enspace\text{for }2\le i\le2001.Prove that ∑i=22001xix1+x2+…+xi−1>1.999\sum_{i=2}^{2001}\frac{x_i}{x_1+x_2+\ldots+x_{i-1}}>1.999.
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