Let n be a positive integer. Find all complex numbers $x_{1}$, $x_{2}$, ..., $x_{n}$ satisfying the following system of equations:

$x_{1}+2x_{2}+...+nx_{n}=0$,
$x_{1}^{2}+2x_{2}^{2}+...+nx_{n}^{2}=0$,
...
$x_{1}^{n}+2x_{2}^{n}+...+nx_{n}^{n}=0$.
1. **Initial Setup and Simplification**:
   We start with the given system of equations:
   \[
   \begin{cases}
   x_1 + 2x_2 + \cdots + nx_n = 0 \\
   x_1^2 + 2x_2^2 + \cdots + nx_n^2 = 0 \\
   \vdots \\
   x_1^n + 2x_2^n + \cdots + nx_n^n = 0
   \end{cases}
   \]
   We assume that not all \( x_i \) are zero. If any \( x_i = 0 \), we can remove it from the system without loss of generality.

2. **Rewriting the System**:
   Let \( k \) be the number of non-zero \( x_i \). We can rewrite the system as:
   \[
   \begin{cases}
   a_1 x_1 + a_2 x_2 + \cdots + a_k x_k = 0 \\
   a_1 x_1^2 + a_2 x_2^2 + \cdots + a_k x_k^2 = 0 \\
   \vdots \\
   a_1 x_1^n + a_2 x_2^n + \cdots + a_k x_k^n = 0
   \end{cases}
   \]
   where \( a_i > 0 \) and \( x_i \neq 0 \).

3. **Grouping Equal Numbers**:
   Suppose we group equal numbers among \( x_1, x_2, \ldots, x_k \). Let \( y_1, y_2, \ldots, y_r \) be the distinct non-zero values, and let \( b_i \) be the sum of the coefficients corresponding to each \( y_i \). The system becomes:
   \[
   \begin{cases}
   b_1 y_1 + b_2 y_2 + \cdots + b_r y_r = 0 \\
   b_1 y_1^2 + b_2 y_2^2 + \cdots + b_r y_r^2 = 0 \\
   \vdots \\
   b_1 y_1^n + b_2 y_2^n + \cdots + b_r y_r^n = 0
   \end{cases}
   \]
   where \( b_i > 0 \) and \( y_i \) are distinct and non-zero.

4. **Matrix Representation and Vandermonde Determinant**:
   Consider the matrix formed by the coefficients of \( b_i \):
   \[
   \begin{pmatrix}
   y_1 & y_2 & \cdots & y_r \\
   y_1^2 & y_2^2 & \cdots & y_r^2 \\
   \vdots & \vdots & \ddots & \vdots \\
   y_1^n & y_2^n & \cdots & y_r^n
   \end{pmatrix}
   \]
   This is a Vandermonde matrix. The determinant of a Vandermonde matrix is given by:
   \[
   \prod_{1 \leq i < j \leq r} (y_j - y_i)
   \]
   Since \( y_i \) are distinct, the determinant is non-zero, which implies that the matrix is non-singular.

5. **Contradiction and Conclusion**:
   The non-singularity of the Vandermonde matrix implies that the only solution to the system is \( b_1 = b_2 = \cdots = b_r = 0 \). However, since \( b_i > 0 \), this leads to a contradiction. Therefore, the assumption that not all \( x_i \) are zero is false.

6. **Final Conclusion**:
   Hence, the only solution to the system is:
   \[
   x_1 = x_2 = \cdots = x_n = 0
   \]

The final answer is \( \boxed{ x_1 = x_2 = \cdots = x_n = 0 } \)