Consider a set of resistors $R_1, R_2, \ldots, R_n>0$ all joined on one end, forming a star-shaped network. Now in turn, for each resistor $R_i$, we let current flow in through $R_i$ and connect the other resistors $R_1, \ldots, R_{i-1}, R_{i+1}, \ldots, R_n$ in parallel, giving a total resistance of
$$\hat R_i = R_i + \Bigl(-R_i^{-1}+\sum_{j=1}^n R_j^{-1}\Bigr)^{-1}.$$
Given $\hat R_0, \hat R_1, \ldots, \hat R_n>0$, how can we find the corresponding $R_1, R_2, \ldots, R_n$? Is a solution always possible?
I am afraid that we could have many solutions. For simplification of notations denote $C_i=1/\hat{R_i}$ , $X_i=1/R_i$ and $S=X_1+\cdots+X_n$ and we have to solve the system for $i=1,\ldots,n:$
$$\frac{1}{C_i}=\frac{1}{X_i}+\frac{1}{S-X_i}$$ or $$X_i^2-X_iS+C_iS=0.$$ Therefore the knowledge of $S$ gives an almost perfect knowledge of $$X_i=\frac{S}{2}\left(1+\epsilon_i\sqrt{1-\frac{4C_i}{S}}\right)$$ where unfortunately $\epsilon_i=\pm 1$ So finally given a choice of $(\epsilon_1,\cdots,\epsilon_n)$ $S$ is a solution of the equation $$S=\frac{n}{2}S+\frac{S}{2}\sum_{i=1}^n\epsilon_i\sqrt{1-\frac{4C_i}{S}}$$ which has to be treated numerically.