7. (CZS 1) Find all real solutions of the system of equations  
$$ \begin{aligned} x_{1}+x_{2}+\cdots+x_{n} & =a, \\ x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2} & =a^{2}, \\ \ldots \cdots \cdots \cdots+x_{n}^{n} & =a^{n} . \end{aligned} $$
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 
\[ k \sigma_{k} = S_{1} \sigma_{k-1} - S_{2} \sigma_{k-2} + \cdots + (-1)^{k} S_{k-1} \sigma_{1} + (-1)^{k-1} S_{k}, \quad k=1,2, \ldots, n \]
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 \( x_{j} \)'s are real.