SOLUTION: Let a, b, c be positive real numbers. Find the minimum value of {{{((1/a)+(1/b)+(1/c))/((1/(a+b))+(1/(a+c))+(1/(c+b)))}}}

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Question 1157229: Let a, b, c be positive real numbers. Find the minimum value of
Found 3 solutions by mananth, ikleyn, math_helper:
Answer by mananth(16946)   (Show Source): You can put this solution on YOUR website!

Numerator =
Denominator =

=
=

=
Denominator =1
Numerator =


Answer by ikleyn(52778)   (Show Source): You can put this solution on YOUR website!
.

The post by @mananth contains elementary algebraic ERRORS,

so it is not a solution.

Do not consider it as a solution.

It is not.



Answer by math_helper(2461)   (Show Source): You can put this solution on YOUR website!



Here is the best I could come up with (see CAVEAT near end):







For a=b=c:







So when the numbers are all equal, we get f(a,b,c) = 2.  I aim to show that 2 is the minimum value.  



Let a be the smallest number (due to symmetry, no loss of generality).



There are two cases to explore, the first case we let one of the other numbers be bigger than a:  



  

  

  



(f=N/D)

N = 



D = 

  =  



N/D = N*(1/D) = 



=  



For any  this last expression is > 2.   EDIT 4/28: to be precise, it is  





The 2nd case is to let both b and c be larger than a.  I am keeping it simple and letting b=c.  This is equivalent to writing , , :



The resulting N*(1/D) expression is

 and this expression is > 2 as well.  EDIT 4/28: the precise value is  





Therefore   



CAVEAT:

This proves f_min=2 holds for three of four scenarios: (1) , (2) , , and (3)  , .   My proof does NOT cover the case , although I'm certain it will hold true.  I leave this case to the student (or motivated tutor).   To handle this case, let , , .  The math will look a lot messier... 



Perhaps another tutor will find a simpler solution(?) 





EDIT 4/28:  The proof for  is very challenging (at least for me).  It has some subtlties that I'm unable to see.  Although not a proof, a SPECIFIC EXAMPLE is easy to illustrate:



Let a=a

    b=2a 

    c=4a





= 

= 

= 

= 

=  > 2



 

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