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a) Suppose that \[|a - b| + |b - c| + |c - a| = 20.\] What is the maximum possible value of $|a - b|$?

 

b) Suppose that \[|a - b| + |b - c| + |c - d| + \dots + |m-n| + |n-o| + \cdots+ |x - y| + |y - z| + |z - a| = 20.\] What is the maximum possible value of $|a - n|$?

Guest Apr 24, 2018
 #1
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Here's the first one

 

Let    a > c > b

 

And  since  a > b, then  l a - b l  =  a - b

And since  c > b, the l b - c l  =  c - b

And since a > c, then l c - a l  = a - c

 

Then  we have that

 

(a - b) + ( c - b) + ( a - c)  = 20  simplify

 

2a  - 2b  = 20       divide through by 2

 

a - b  =  10

 

And this is the max value for  l a - b l

 

 

 

cool cool cool

CPhill  Apr 24, 2018

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