An algebra problem by Sourabh Jangid

Algebra Level 5

{x=y2116+z2116y=z2125+x2125z=x2136+y2136 \begin{cases} x = \sqrt{y^2 - \frac1{16} } + \sqrt{z^2 - \frac1{16}} \\ y = \sqrt{z^2 - \frac1{25} } + \sqrt{x^2 - \frac1{25}} \\ z = \sqrt{x^2 - \frac1{36} } + \sqrt{y^2 - \frac1{36}} \\ \end{cases}

Let x,yx,y and zz be real numbers satisfying the system of equations above. If the value of x+y+zx+y+z can be expressed as mn \dfrac m{\sqrt n} , where mm and nn are positive integers with nn square-free, find m+nm+n.


This problem is from AIME 2006.

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