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Wednesday, January 15, 2014

Infimum of abc

Find the largest number M such that for any given distinct number a,b,c such that 0 < a,b,c < 1 we always have: \frac{(a+b)(b+c)(c+a)}{|(a-b)(b-c)(c-a)|} > M

Solution

Clearly M=1 satisfies the condition. Now we prove that we can get the LHS arbitrarily close to 1.

Let a be an arbitrary number < 1, b = a/(1+x) and c=a/(1+x+yx) for some really large numbers x,y. The LHS now becomes: \frac{(2+x)(2+(y+1)x)(2+(y+2)x)}{y(y+1)x^3} =(\frac{2}{x}+1)(\frac{2}{x(y+1)} + 1)(\frac{2}{xy} + \frac{2}{y} + 1) We can set x,y large enough that the expression is as close as needed to 1.

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