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max of (1/2001 \sum x_n^2)-(1/2001 \sum x_n)^2 , n=1 to 2001

Source: Israel Grosman Memorial Mathematical Olympiad 2001 p2

February 15, 2020
Suminequalitiesalgebra

Problem Statement

If x1,x2,...,x2001x_1,x_2,...,x_{2001} are real numbers with 0≤xn≤10 \le x_n \le 1 for n=1,2,...,2001n = 1,2,...,2001, find the maximum value of (12001∑n=12001xn2)−(12001∑n=12001xn)2\left(\frac{1}{2001}\sum_{n=1}^{2001}x_n^2\right)-\left(\frac{1}{2001}\sum_{n=1}^{2001}x_n\right)^2 Where is this maximum attained?
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