MathDB

Given a sequence

s

consisting of digits

0

and

1

. For any positive integer

k

, define

v_k

the maximum number of ways in any sequence of length

k

that several consecutive digits can be identified, forming the sequence

s

. (For example, if

s=0110

, then

v_7=v_8=2

, because in sequences

0110110

and

01101100

one can find consecutive digits

0110

in two places, and three pairs of

0110

cannot meet in a sequence of length

7

8

.) It is known that

v_n<v_{n+1}<v_{n+2}

for some positive integer

n

. Prove that in the sequence

s

, all the numbers are the same.
A. Golovanov

Problem Statement