I aim to study the following system of equations with respect to $(x_{1}, \dots, x_{m}, y_{1}, \dots, y_{m})$
$$ \left\{ \begin{array}{c} y_1x_1+y_2x_2+y_3x_3 + \dots + y_mx_m=c_1 \\ y_1x_1^2+y_2x_2^2+y_3x_3^2 + \dots + y_mx_m^2=c_2\\ y_1x_1^3+y_2x_2^3+y_3x_3^3 + \dots + y_mx_m^3=c_3\\ \vdots\\ y_1x_1^n+y_2x_2^n+y_3x_3^n + \dots + y_mx_m^n=c_n\\ \end{array} \right. $$
with the constraint $y_{1} + y_{2} + \dots + y_{m} = 1$. We assume that $n=2m-1$.
Is there any way to solve it? I do not know how to start.