For $i=1,2,\ldots,n$, let $d_i$ be a sequence of positive integers and $r_i$ a sequence of positive real numbers with $2\leq d_i\leq n-1$ that satisfy the following system of equations:
$d_i=\sum_{j=1}^n \frac{r_ir_j}{1+r_ir_j}$
for each $i=1,2\ldots,n$.
Question: Can we find a qualitative bound on $r_i$ in terms of $n$? I don't care if it's is a terrible bound, I just want any bound in terms of $n$.
Everything I've tried thus far seems to give me lower bounds on $r_i$. I've considered fudged cases (i.e. not requiring $d_i$ to be integers) of taking part of the $r_i$ to be some number $L$ and the other part to be $1/L$ in hopes of breaking any bound, but one can show that $L$ will be of order $\sqrt{n}$ thereby being bounded for a fixed $n$.
If we suppose wlog that
(*) $r_1\leq r_2\leq\cdots\leq r_n$,
then a primitive bound on $r_i$ would be:
$\left(\frac{r_1}{1+r_n^2}\right)r_i\leq d_i\leq \left(\frac{r_n}{1+r_1^2}\right)r_i$
but again this is not useful as this doesn't give any information on $r_n$.
Any help with this would be greatly appreciated!