- #1

Bernie Hunt

- 19

- 0

The Problem;

If a>0, b>0, c>0 and a + b > c, b + c > a, c + a > b then ( a + b + c )^2 <= 4( ab + bc + ca ).

Hints: Start with ( a + b + c )^2 and establish the inequality ( a + b + c )^2 <= 4( ab + bc + ca ). Us the inequality fact: if x < y then xy < y^2 if y > 0.

I haven’t make much progress. I’ve been trying;

( a + b + c )^2 = a^2 + b^2 + c^a + 2ab + 2bc + 2ca

But I haven’t been able to figure out a way to get the right side manipulated. I tried a couple of substitutions, but couldn’t justify a<b or such to be able to use the hint.

Can anyone give me a push in the right direction?

Thanks,

Bernie

If a>0, b>0, c>0 and a + b > c, b + c > a, c + a > b then ( a + b + c )^2 <= 4( ab + bc + ca ).

Hints: Start with ( a + b + c )^2 and establish the inequality ( a + b + c )^2 <= 4( ab + bc + ca ). Us the inequality fact: if x < y then xy < y^2 if y > 0.

I haven’t make much progress. I’ve been trying;

( a + b + c )^2 = a^2 + b^2 + c^a + 2ab + 2bc + 2ca

But I haven’t been able to figure out a way to get the right side manipulated. I tried a couple of substitutions, but couldn’t justify a<b or such to be able to use the hint.

Can anyone give me a push in the right direction?

Thanks,

Bernie

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