- #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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