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MC-IV week 1 homework help

 
 
ChenWilliam的头像
MC-IV week 1 homework help
ChenWilliam 发表于 2018年06月15日 Friday 19:13
 

Can someone give me a hint on problem #5? I used AM-GM on both sides and reached the same expression on both sides. I know I need to find something in middle but so far no luck.

 
ProfessorAreteem的头像
Re: MC-IV week 1 homework help
ProfessorAreteem 发表于 2018年08月19日 Sunday 15:58
 

Have you tried Rearrangement Inequality?

Note: the problem in question is the following:

Let $a_i>0 (i=1,2,\ldots,n)$, and real numbers $x,y$ satisfy $xy>0$. Show
$$
\frac{a_1^x}{a_2^y}+\frac{a_2^x}{a_3^y}+\cdots+\frac{a_{n-1}^x}{a_n^y}+\frac{a_n^x}{a_1^y} \geq \sum_{i=1}^n a_i^{x-y}.
$$

(Edited by Areteem Institute based on comment from William Chen- original submission Friday, June 15, 2018, 11:56 PM)

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Re: MC-IV week 1 homework help
ChenWilliam 发表于 2018年06月17日 Sunday 16:35
 
Is there a typo in the problem? I think the right hand side should be 

Should $\sum_{i=1}^n a_n^{x-y}$ be $\sum_{i=1}^n a_i^{x-y}$