- #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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