If I'm not mistaken, this equation has... a lot of solutions? Let $ (x,y,z)$ be any positive reals such that $ x + y + z = 1$, and let $ k = \frac {\frac {xy}{2005y + 2004x} + \frac {yz}{2004z + 2003y} + \frac {zx}{2003x + 2005z}}{\frac {x^2 + y^2 + z^2}{2005^2 + 2004^2 + 2003^2}}$. Then $ (kx, ky, kz)$ seems to satisfy the equation. Conversely, for any solution $ (x,y,z)$ can be written as $ (ka, kb, kc)$, where $ a + b + c = 1$ (simply let $ k = (x + y + z), a = \frac {x}{k}, b = \frac {y}{k}, c = \frac {z}{k}$.) Aside from that, setting $ a = \frac {x}{2005}$, $ b = \frac {y}{2004}$, and $ c = \frac {z}{2003}$ seemed to make things a bit nicer...

Thank you.