Irrational Equations

Algebra Level 5

We are given the following system of equations

{11+2x2+11+2y2=21+2xy,x(12x)+y(12y)=29.\begin{cases} \frac{1}{\sqrt{1+2x^2}}+\frac{1}{\sqrt{1+2y^2}}=\frac{2}{\sqrt{1+2xy}}, \\ \sqrt{x(1-2x)}+\sqrt{y(1-2y)}=\frac{2}{9}. \end{cases}

If xx can be expressed in the form a±bc,\frac{a\pm\sqrt{b}}{c}, where a,ba,b and cc are positive integers and bb is not divisible by the square of any prime, find the value of a+b+c.a+b+c.

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