Solution:

Let $x_{0}, \ldots, x_{199}$ be variables. Assign the value of $+1$ or $-1$ to $x_{i}$ depending on whether the $(i+1)$st segment of the larger circle (counting counterclockwise) is black or white, respectively. Similarly, assign the value of $+1$ or $-1$ to the variable $y_{i}$ depending on whether the $(i+1)$th segment of the smaller circle is black or white. We can now restate the problem in the following equivalent way: show that
$$S_{j} = \sum_{i=1}^{200} x_{i} y_{i+j} \geq 0$$
for some $j = 0, \ldots, 199$. Here the subscript $i+j$ is understood modulo $200$.

Now observe that $y_{0} + \cdots + y_{199} = 0$ and thus
$$S_{0} + \cdots + S_{199} = \sum_{i=0}^{199} x_{i} (y_{0} + \cdots + y_{199}) = 0.$$
Thus $S_{j} \geq 0$ for some $j = 0, \ldots, 199$, as claimed.