How to solve equations of algebra? Let $a_i>0, b_i>0$ ($i=1,2,\ldots,N$). How to prove that there exist unique $x_i>0$ ($i=1,2,\ldots,N$) such that
$$a_ix_i^{b_i}+x_1+x_2+\cdots+x_N=1,\;\;i=1,2,\ldots,N.$$
Thank you.

Replace $x_1+\ldots+x_N$ in the equation by $S$. A solution to the problem must have $0<S<1$, since $0<a_ix_i^{b_i}=1-S$. Then 
$$
x_i=(\frac{1-S}{a_i})^{1/b_i}=f_i(S).
$$
$f_i(S)$ are continuous functions from $[0,1]$ to $\mathbb{R^{+}}$, so the same is true for the sum $F(S)=\sum_{i=1}^N f_i(S)$. $F(S)$ is strictly monotonly decreasing with $F(0)=\sum\frac{1}{a_i}^{1/b_i}>0$ and $F(1)=0$. Thus there is a unique point where $F(S)=S$. Now you should be able to conclude.
