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Part of 2011 ISI B.Stat Entrance Exam

Problems(1)

An easy ineq; ISI BS 2011, P1

Source:

3/31/2013
Let x1,x2,⋯ ,xnx_1, x_2, \cdots , x_n be positive reals with x1+x2+⋯+xn=1x_1+x_2+\cdots+x_n=1. Then show that ∑i=1nxi2−xi≥n2n−1\sum_{i=1}^n \frac{x_i}{2-x_i} \ge \frac{n}{2n-1}
calculusinequalitiesrearrangement inequalityn-variable inequality