How Can we proof that equation (1)  have 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*}