2013 Putnam 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 $N(w) = N$.

数字 $0$ 和 $1$ 的有限集合写在一个圆圈周围。

长度为 $L \geq 0$ 的 *arc* 由圆周围的 $L$ 个连续数字组成。对于每个弧$w$，让$Z(w)$和$N(w)$分别表示$w$中$0$的数量和$w$中$1$的数量。

假设 $\left| Z(w) - Z(w') \right| \leq 1$ 对于任意两个相同长度的弧 $w, w'$。假设某些弧 $w_1,\dots,w_k$ 具有以下属性

$$

Z = \frac{1}{k} \sum_{j=1}^k Z(w_j) \text{ 和 }

N = \frac{1}{k} \sum_{j=1}^k N(w_j)

$$

都是整数。证明存在 $w$ 且 $Z(w) = Z$

且 $N(w) = N$。