My prof. gave this problem as a bonus in an exam, and I couldn't figure out a solution. Some hints or a general method of solving it would be very nice. Given the following polynomial: $$P(x)=a_{3}x^{3}+a_{5}x^{5}+a_{7}x^{7}+\dots+a_{199}x^{199}+a_{201}x^{201}$$
and knowing its solutions, provided in the following plot
(where the blue/red dots represent $\mathrm{Re[x]}/\mathrm{Im[x]}$) find the coefficients $a_{2k+1}$. The problem also stated that $a_{2k+1}\in\mathbb{R}_{>0}$ and $\lim_{k\rightarrow\infty}a_{2k+1}=1$ (in the exam we also had a very long list of all the numerical solutions for $P(x)$).
Given the roots of a polynomial, call them $r_1, r_2, r_3, \ldots r_{201}$, then the coefficients are given by the elementary symmetric functions on the roots.
If you are not familiar with these, read this article on Wikipedia
So, $a_{2k+1} = \dfrac{e_{201-2k -1}(r_1, r_2, \ldots r_{201})}{a_{201}}$
This gives a "general" solution. Unless he/she gave you what $a_{201}$ is, I am not sure of a good method for getting it (since it actually can be whatever you want; the other coefficients will then depend on it, as we divide by $a_{201}$. This perhaps indicates that $a_{201} = 1$ could be a good choice!).
If you have the roots given to you, do some sort of thing like this: $$\prod_{i =1}^{201}(x-r_i)$$ on a computer. I am assuming that this is a numerical methods class and you have some computer ability. It should spit out what your polynomial is.