- #1
ehrenfest
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Homework Statement
Let a,b,c be the side lengths of a triangle with semiperimeter 1. Prove that
[tex]1 < ab + bc + ca - abc \leq 28/27[/tex]
Homework Equations
the AM-GM inequality:
If x_1,...,x_n are nonnegative real numbers, then
[tex]\frac{\sum x_i}{n} \leq \left( \Pi x_i\right)^{1/n}[/tex]
The Attempt at a Solution
I can get "close" to the second inequality. Applying AM-GM to a+b-c,a-b+c,-a+b+c gives
[tex]8/27 \geq a^3 +b^3 +c^3 -2abc +a^2b+ab^2+ac^2 +a^2 c+bc^2 +b^2 c [/tex]
Applying AM-GM to a,b,c gives
[tex]8/27 \geq abc [/tex]
and then we can add the two inequalities to obtain
[tex]16/27 \geq a^3 +b^3 +c^3 -abc +a^2b+ab^2+ac^2 +a^2 c+bc^2 +b^2 c [/tex]
Ok maybe that's not really close...
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