SILK-ROAD 2021 Problem 1

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 $7$ or $8$ .) 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