My Math Forum Inequality challenge

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 March 30th, 2014, 11:46 AM #1 Senior Member   Joined: Mar 2012 From: Belgium Posts: 654 Thanks: 11 Inequality challenge Given some real numbers $x_1 \;,\; x_2 \;,\; x_3 \;,\; ... \;,\; x_n \; > \; -1$ with $x_1^3 \;+\; x_2^3 \;+\; x_3^3 \;+\; ... \;+\; x_n^3 \;= \;0$ prove that $x_1 \;+\; x_2 \;+\; x_3 \;+\; ... \;+\; x_n \; \leq \; \frac{n}{3}$ Thanks from fahad nasir
 March 31st, 2014, 08:41 AM #2 Senior Member   Joined: Mar 2012 From: Belgium Posts: 654 Thanks: 11 Hint: Consider the polynomial $x^3 - \frac {3}{4}x+\frac {1}{4}$
 March 31st, 2014, 10:03 AM #3 Global Moderator   Joined: Dec 2006 Posts: 20,966 Thanks: 2216 It's easy to show that (4/3)x³ + 1/3 ≥ x for x ≥ -1. Applying that to each numbered x variable and adding gives the desired inequality.
 March 31st, 2014, 10:38 AM #4 Senior Member   Joined: Mar 2012 From: Belgium Posts: 654 Thanks: 11 Well done!

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