bridge hands and math and computer science and programming and puzzles and etc and etc.
Wednesday, October 3, 2018
Yet another inequality
x1,x2,…,xn are n positive real numbers with sum S, n>1.
Show that
n∑i=1xiS−xi≥nn−1
Solution [Click here to expand/collapse]
Using AM >= GM we have that
∑yi∑1yi≥n2
Applying this to (wlog, assuming S=1)
(n∑i=111−xi)(n∑i=1(1−xi))≥n2
i.e
n∑i=111−xi≥n2n−1
And so
n∑i=1(11−xi−1)≥n2n−1−n
Which gives us what we want to prove.
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