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Weird Inequality with 2019 variables

Source: 2019 China Second Round P2

September 8, 2019
algebracombinatoricsChina second roundinequalitiesn-variable inequality

Problem Statement

Let a1,a2,⋯ ,ana_1,a_2,\cdots,a_n be integers such that 1=a1≤a2≤⋯≤a2019=991=a_1\le a_2\le \cdots\le a_{2019}=99. Find the minimum f0f_0 of the expression f=(a12+a22+⋯+a20192)−(a1a3+a2a4+⋯+a2017a2019),f=(a_1^2+a_2^2+\cdots+a_{2019}^2)-(a_1a_3+a_2a_4+\cdots+a_{2017}a_{2019}), and determine the number of sequences (a1,a2,⋯ ,an)(a_1,a_2,\cdots,a_n) such that f=f0f=f_0.
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