Is there a closed form solution for these equations

Question

$\theta_i (i=1,\ldots,N)$ are real numbers and we have $$ \sum_{i=1}^N \theta_i = 1 $$ For any $i\neq j$, $$ \sum_{w\in W} \frac{f_i(w)}{\sum_{k=1}^N \theta_k f_k(w)} = \sum_{w\in W} \frac{f_j(w)}{\sum_{k=1}^N \theta_k f_k(w)} $$

Here $W$ is a set, and for any $i$, $f_i()$ is a function that maps an element in $W$ to a scalar.

I guess there should be a closed form solution for $\theta_i (i=1,\ldots,N)$ (in terms of $f_i$ and $W$) but couldn't figure it out. Thank you!

Update

The equations above are what I get by applying the Karush–Kuhn–Tucker conditions to the following optimization problem:

Maximize $$ \prod_{w\in W} \sum_{k=1}^N \theta_k f_k(w) $$ subject to $$ \sum_{i=1}^N \theta_i = 1\\ \forall i, \theta_i\geq 0 $$

Answer 1

I looked at $N=2$. To simplify notation, I took $\theta_1=a$, $\theta_2=b$, $W=\{{r,s\}}$, $f_1(r)=u$, $f_1(s)=v$, $f_2(r)=w$, $f_2(s)=x$, and I got $$a={ux+vw-2wx\over2(u-w)(x-v)}$$ with something similar for $b$.