# Finding the range of solutions

Say I have the following equations (I am given n, c, and d): $$\frac{1}{n} \sum _{i=1}^{n}f(x_{i} ,y_{i} ) =c$$ $$\frac{1}{n} \sum _{i=1}^{n}g(x_{i} ,y_{i} ) =d$$ $$0<x_{i} ,y_{i} <1$$

And I need to find: $$S=\frac{1}{n} \sum _{i=1}^{n}h(x_{i} ,y_{i} ) =?$$ where,

$$f(x_{i} ,y_{i} )=2(-x_{i} ^{4} -y_{i} ^{4} +2x_{i} ^{3} +2y_{i} ^{3} -2x_{i} ^{2} -2y_{i} ^{2} +x_{i} +y_{i} )$$ $$g(x_{i} ,y_{i} )=2(2x_{i} y_{i} ^{2} +2x_{i} ^{2} y_{i} -2x_{i} ^{2} y_{i} ^{2} -4x_{i} y_{i} +x_{i} +y_{i} )$$ $$h(x_{i} ,y_{i} )=\frac{(x_{i} -y_{i} )^{2}}{(x_{i} +y_{i} )(2-x_{i} -y_{i})}$$

Obviously, there are generally infinitely many solutions for the first two equations (two equations and 2n variables), but perhaps all those solutions result in a "narrow" range for S, for any n, c and d, and therefore I can say something meaningful about S.

Is there any way to know this?

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Thanks, I provided the functions now. –  Omri Dec 17 '11 at 11:05