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\sum x_i ^2 \le -nmM if \sum x_i = 0 and x_i \in [m,M]

Source: Norwegian Mathematical Olympiad 2003 - Abel Competition p1b

February 21, 2020
Suminequalitiesalgebra

Problem Statement

Let x1,x2,...,xnx_1,x_2,...,x_n be real numbers in an interval [m,M][m,M] such that ∑i=1nxi=0\sum_{i=1}^n x_i = 0. Show that ∑i=1nxi2≤−nmM\sum_{i=1}^n x_i ^2 \le -nmM
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