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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