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Sum of x_1=1 inequality

Source: Baltic Way 2002

November 13, 2010
inequalitiesinequalities proposed

Problem Statement

Let nn be a positive integer. Prove that ∑i=1nxi(1−xi)2≤(1−1n)2\sum_{i=1}^nx_i(1-x_i)^2\le\left(1-\frac{1}{n}\right)^2 for all nonnegative real numbers x1,x2,…,xnx_1,x_2,\ldots ,x_n such that x1+x2+…xn=1x_1+x_2+\ldots x_n=1.
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