blackdivcp

blackdivcp

Answered

2022-10-12

Prove that 1 a + b + 1 + 1 b + c + 1 + 1 c + a + 1 1
If a b c = 1 then
1 a + b + 1 + 1 b + c + 1 + 1 c + a + 1 1
I have tried AM-GM and C-S and can't seem to find a solution. What is the best way to prove it?

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Recalculate according to your conditions!

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wespee0

wespee0

Expert

2022-10-13Added 15 answers

Also we can use the following reasoning.
For positives a, b and c let a = x 3 , b = y 3 and c = z 3
Hence, c y c 1 a + b + 1 = c y c 1 x 3 + y 3 + x y z c y c 1 x 2 y + y 2 x + x y z = c y c z x + y + z = 1,
but I think the first way is better.

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Krish Logan

Krish Logan

Expert

2022-10-14Added 4 answers

It's wrong, of course.
For positives a, b and c it's c y c ( a 2 b + a 2 c 2 a ) 0, which is Muirhead because ( 2 , 1 , 0 ) ( 5 3 , 2 3 , 2 3 ) .

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