a difficult inequality

skywalkerJ.L.

Prove that for $n$ distinct positive integers $a_1,a_2,\ldots,a_n$ ,the following inequality holds:
${a_1}^3+{a_2}^3+...{a_n}^3\ge (a_1+a_2+\ldots+a_n)^2$

**Posted:** Thu Nov 05, 2009 8:26 am

leshik · Poincare Conjecture Offline Joined: 03 Aug 2009 Posts: 156

Simple proof can be based on induction. To prove the step from $n-1$ to $n$ it is enough to show that $a^2_n\ge a_n +2(\sum\limits_{i=1}^{n-1} a_i)$ . Now just use obvious inequalities $a_{n-i}\le a_n -i$ and $a_n\ge n$ Wink

**Posted:** Thu Nov 05, 2009 8:39 am

MeKnowsNothing · Riemann Hypothesis Offline Joined: 27 Feb 2007 Posts: 370

Am I right that because of Newton's sums, one can reduce this to three variable inequality?

**Posted:** Thu Nov 05, 2009 10:38 am

skywalkerJ.L.

**Posted:** Sat Nov 07, 2009 2:15 am

MeKnowsNothing · Riemann Hypothesis Offline Joined: 27 Feb 2007 Posts: 370

**Posted:** Sat Nov 07, 2009 5:33 am