How can we prove that equation (1) has solutions for every $p$. I mean, is there an analytic method that can be used to show that there exist solutions for every $p$ for this nonlinear equation:

\begin{equation} \left\{ \begin{array}{ccc} x_1^2 x_2 \cdots x_{p-1} x_p+x_1 &=& 1 \, ,\\ &&\\ x_1 x_2^2 x_3 \cdots x_{p-1} x_p+x_1 x_2 &=& 1 \, ,\\ &&\\ x_1 x_2 x_3^2 x_4 \cdots x_{p-1} x_p+x_1 x_2 x_3&=& 1\, , \\ &&\\ \vdots & \vdots & \vdots \\ &&\\ x_1 x_2 x_3 x_4 \cdots x_{p-1} x_p^2+x_1 x_2 \cdots x_p&=& 1\, . \end{array} \right. \end{equation}

For example, let ($p=3$). Then we have:

\begin{equation} \left\{ \begin{array}{ccc} x_1^2 x_2 x_3+x_1 &=& 1 \, ,\\ &&\\ x_1 x_2^2 x_3 +x_1 x_2 &=& 1 \, ,\\ &&\\ x_1 x_2 x_3^2 +x_1 x_2 x_3&=& 1\, . \end{array} \right. \end{equation}

So is there an analytic method for existence of solution for equation (2). For this example I used MAPLE software and got these solutions:

\begin{equation*} \left\{ \begin{array}{ccc} a &=& 0.658418845314095780 \, ,\\ &&\\ b &=& 0.849466898144101812 \, ,\\ &&\\ c&=& 0.927561975482924960 \, . \end{array} \right. \end{equation*}