11. (ROM 6) Find all solutions $\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ of the equation $1+\frac{1}{x_{1}}+\frac{x_{1}+1}{x_{1} x_{2}}+\frac{\left(x_{1}+1\right)\left(x_{2}+1\right)}{x_{1} x_{2} x_{3}}+\cdots+\frac{\left(x_{1}+1\right) \cdots\left(x_{n-1}+1\right)}{x_{1} x_{2} \cdots x_{n}}=0$.
11. Introducing $y_{i}=\frac{1}{x_{i}}$, we transform our equation to  $$ \begin{aligned} 0 & =1+y_{1}+\left(1+y_{1}\right) y_{2}+\cdots+\left(1+y_{1}\right) \cdots\left(1+y_{n-1}\right) y_{n} \\ & =\left(1+y_{1}\right)\left(1+y_{2}\right) \cdots\left(1+y_{n}\right) \end{aligned} $$  The solutions are $n$-tuples $\left(y_{1}, \ldots, y_{n}\right)$ with $y_{i} \neq 0$ for all $i$ and $y_{j}=-1$ for at least one index $j$. Returning to $x_{i}$, we conclude that the solutions are all the $n$-tuples $\left(x_{1}, \ldots, x_{n}\right)$ with $x_{i} \neq 0$ for all $i$, and $x_{j}=-1$ for at least one index $j$.