MathDB

4

Part of 2009 South africa National Olympiad

Problems(1)

Inequality between product, sum and sum of reciprocals

Source: South African MO 2009 Q4

5/26/2012
Let x1,x2,…,xnx_1,x_2,\dots,x_n be a finite sequence of real numbersm mwhere 0<xi<10<x_i<1 for all i=1,2,…,ni=1,2,\dots,n. Put P=x1x2⋯xnP=x_1x_2\cdots x_n, S=x1+x2+⋯+xnS=x_1+x_2+\cdots+x_n and T=1x1+1x2+⋯+1xnT=\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}. Prove that T−S1−P>2.\frac{T-S}{1-P}>2.
inequalitiesinequalities unsolved