Min Value of 1/6 (a^3 + b^3 + c^3 - 3abc) for Distinct Ints

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Homework Statement


If a , b ,c r distinct +ve integers such that ab + bc + ca is greater than equal to 107 , then find the minimum value of 1/6 ( a^3 + b^3 + c^3 - 3abc )


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The Attempt at a Solution


tried usin AM GM equality but i m confused which nos. shud i use??
 
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This smells very much like Lagrange multipliers
 
got the ans finally using AM GM only...:biggrin:
 
NEILS BOHR said:
got the ans finally using AM GM only...:biggrin:
Can you show me?... i got stuck too. XD
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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