Putnam 2013 A4

A finite collection of digits $0$ and $1$ is written around a circle. An arc of length $L \geq 0$ consists of $L$ consecutive digits around the circle. For each arc $w$, let $Z(w)$ and $N(w)$ denote the number of $0$'s in $w$ and the number of $1$'s in $w$, respectively. Assume that $\left| Z(w) - Z(w') \right| \leq 1$ for any two arcs $w, w'$ of the same length. Suppose that some arcs $w_1,\dots,w_k$ have the property that $$Z = \frac{1}{k} \sum_{j=1}^k Z(w_j) \text{ and } N = \frac{1}{k} \sum_{j=1}^k N(w_j)$$ are both integers. Prove that there exists an arc $w$ with $Z(w) = Z$ and $N(w) = N$.