VMO 2005

From the condition, we can see that $x+y=3(\sqrt{x+1}+\sqrt{y+2})$ Now if we set $\sqrt{x+1}=t$ and $\sqrt{y+2}=u$ then we'll have this following system $\begin{cases} u^2+t^2-3=3(u+v)\\u^2+v^2-3=A\end{cases}$ $\Leftrightarrow\begin{cases} u+v=\frac{A}{3}\\ uv=\frac{1}{2}\bigg(\frac{A^2}{9}-A-3\bigg)\end{cases}$ So $u,v$ are the roots of $X^2-\frac{A}{3}X+\frac{1}{2}\bigg(\frac{A^2}{9}-A-3\bigg)=0 \ (*)$ In order for $(*)$ to have 2 non-negative roots $\begin{cases} \Delta\geq 0\\ x_1x_2\geq 0\\ x_1+x_2\geq 0\end{cases}$ $\Leftrightarrow\begin{cases}\frac{-A^2}{9}+2A+6\geq 0\\ \frac{A^2}{9}-A-3\geq 0\\ \frac{A}{3}\geq 0\end{cases}$ $\Leftrightarrow \frac{9+3\sqrt{21}}{2}\leq A\leq 9+3\sqrt{15}$ Therefore the sum is $\approx 31.992$