Solution to $x_i=a(1+\sum_{i\ne j}\frac{b_ix_j}{1+b_ix_j})$

Question

I'm dealing with a set of variables $x_i$ and following equations:

$\forall i$ $x_i=a(1+\sum_{i\ne j}\frac{b_ix_j}{1+b_ix_j})$

Wolfram mathematica's NSolve could help, but it's not very accurate and is very limited in capacity for the number of $x_i$s, like 8.

I'm aiming for 20-100 $x_i$s so, therefore, looking for other ways like an algorithm to find $x_i$s. Any suggestion?

Answer 1

Adding and subtracting $\dfrac{x_i}{1+x_i}$ inside the parenthesis gives

$$x_i = a(1 + \sum_{j}\frac{x_j}{1+x_j} - \frac{x_i}{1+x_i})$$

The sum does not depend on $i$; it is a constant. Can you continue?