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Inequality between product, sum and sum of reciprocals

Source: South African MO 2009 Q4

May 26, 2012

inequalitiesinequalities unsolved

Problem Statement

Let

x_1,x_2,\dots,x_n

be a finite sequence of real numbersm mwhere

0<x_i<1

for all

i=1,2,\dots,n

. Put

P=x_1x_2\cdots x_n

S=x_1+x_2+\cdots+x_n

and

T=\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}

. Prove that

\frac{T-S}{1-P}>2.