Number of triangles ABC if tan A = x, tan B = x + 1,and tan C = 1- x is_____. 

$\mathrm{△}\mathrm{ABC},\tan \mathrm{A}+\tan \mathrm{B}+\tan \mathrm{C}=\tan \mathrm{A} \tan \mathrm{B} \tan \mathrm{C}$
$\Rightarrow \mathrm{x}+\mathrm{x}+1+1-\mathrm{x}=\mathrm{x}(1+\mathrm{x})(1-\mathrm{x})$
or  $2+\mathrm{x}=\mathrm{x}-{\mathrm{x}}^3$
or  ${\mathrm{x}}^3=-2\Rightarrow \mathrm{x}=-2^{1/3}$
$\begin{array}{l}\Rightarrow \tan \mathrm{A}=\mathrm{x}<0\Rightarrow \mathrm{A}\text{ is obtuse }\\ \Rightarrow \tan \mathrm{B}=\mathrm{x}+1=1-2^{1/3}<0\end{array}$
Hence, A and B are obtuse, which is not possible in a triangle.
Hence, no such triangle can exist.