\r
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document.write( "Here is the best I could come up with (see CAVEAT near end):\r
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document.write( "
\r
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document.write( "For a=b=c:\r
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document.write( "
\r
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document.write( "So when the numbers are all equal, we get f(a,b,c) = 2. I aim to show that 2 is the minimum value. \r
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document.write( "Let a be the smallest number (due to symmetry, no loss of generality).\r
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document.write( "There are two cases to explore, the first case we let one of the other numbers be bigger than a: \r
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document.write( " 
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document.write( " 
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document.write( " 
\r
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document.write( "(f=N/D)
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document.write( "N = 
\r
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document.write( "D = 
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document.write( " = 
\r
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document.write( "N/D = N*(1/D) = 
\r
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document.write( "\n" );
document.write( "= 
\r
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document.write( "For any 
this last expression is > 2. EDIT 4/28: to be precise, it is 
\r
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document.write( "
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document.write( "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 
, 
, 
:\r
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document.write( "The resulting N*(1/D) expression is
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document.write( "
and this expression is > 2 as well. EDIT 4/28: the precise value is 
\r
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document.write( "
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document.write( "Therefore 
\r
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document.write( "CAVEAT:
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document.write( "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... \r
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document.write( "Perhaps another tutor will find a simpler solution(?) \r
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document.write( "
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document.write( "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:\r
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document.write( "Let a=a
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document.write( " b=2a
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document.write( " c=4a\r
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document.write( "\n" );
document.write( "
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document.write( "= 
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document.write( "= 
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document.write( "= 
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document.write( "= 
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document.write( "= 
> 2\r
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document.write( "
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document.write( "![\"\"](\"/images/stars/stars-13.gif\")
