If $x,y,z>0$ and $x(1-y)>{1\over 4}$ , $y(1-z)>{1\over 4}$ , $z(1-x)>{1\over 4}$ then find the number of order triplets (x,y,z) satisfying the above inequaltiy.
I am stuck as I know how to deal with inequalities in two variables graphically or otherwise but I could not expand that approach to work with this question.
Could someone please help with how to proceed in this ?
Thanks.
One not so standard way I could think of was since the inequalities are symmetric so x=y=z should be satisfied by its solutions , putting this I got that there are no solution, and the answer of the question is also 0 . But this does not seem to be a nice solution
We can get an inequality of two variables, which you know, by the following way.
By AM-GM $$\prod_{cyc}x(1-y)=\prod_{cyc}x(1-x)\leq\prod_{cyc}\left(\frac{x+1-x}{2}\right)^2=\frac{1}{64}.$$ Can you end it now?