8.1. For which $n$ do there exist positive numbers $x_{1}, \ldots, x_{n}$, satisfying the system of equations

$$
x_{1}+x_{2}+\ldots+x_{n}=3, \quad \frac{1}{x_{1}}+\frac{1}{x_{2}}+\ldots+\frac{1}{x_{n}}=3 ?
$$
8.1. Using the inequality $x_{i}+1 / x_{i} \geqslant 2$, we get $3+3=\left(x_{1}+1 / x_{1}\right)+\ldots+\left(x_{n}+1 / x_{n}\right) \geqslant 2 n$, i.e., $n \leqslant 3$. For $n=3$, we obtain the solution $x_{1}=x_{2}=x_{3}=1$. For $n=2$, we obtain the solution $x_{1}=\frac{3+\sqrt{5}}{2}, x_{2}=\frac{3-\sqrt{5}}{2}$. For $n=1$, the system is inconsistent.