MathDB

A finite collection of digits

0

and

1

is written around a circle. An arc of length

L\ge 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

|Z(w)-Z(w')|\le 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=\frac1k\sum_{j=1}^kZ(w_j)\text{ and }N=\frac1k\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.

Problem Statement