7. (CZS 1) Find all real solutions of the system of equations
$$
\begin{aligned}
x_{1}+x_{2}+\cdots+x_{n} & =a, \\
x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2} & =\alpha^{2} \\
\cdots \cdots \cdots \cdots+ & \cdots \\
x_{1}^{n}+x_{2}^{n}+\cdots+x_{n}^{n} & =\alpha^{n} .
\end{aligned}
$$
7. Let $S_{k}=x_{1}^{k}+x_{2}^{k}+\cdots+x_{n}^{k}$ and let $\sigma_{k}, k=1,2, \ldots, n$ denote the $k$ th elementary symmetric polynomial in $x_{1}, \ldots, x_{n}$. The given system can be written as $S_{k}=a^{k}, k=1, \ldots, n$. Using Newton's formulas
$$
k \sigma_{k}=S_{1} \sigma_{k-1}-S_{2} \sigma_{k-2}+\cdots+(-1)^{k} S_{k-1} \sigma_{1}+(-1)^{k-1} S_{k}, \quad k=1,2, \ldots, n \text {, }
$$
the system easily leads to $\sigma_{1}=a$ and $\sigma_{k}=0$ for $k=2, \ldots, n$. By Vieta's formulas, $x_{1}, x_{2}, \ldots, x_{n}$ are the roots of the polynomial $x^{n}-a x^{n-1}$, i.e., $a, 0,0, \ldots, 0$ in some order.
Remark. This solution does not use the assumption that the $x_{f}$ 's are real.