Question 20
A sequence $s$ consisting of zeroes and ones is given. For each positive integer $k$ we define $v_k$ as the maximum number of ways to find consecutive digits forming the sequence $s$ in a sequence of $k$ digits. (For example, if $s = 0110$, then $v_7 = v_8 = 2$, since in the sequences 0110110 and 01101100 consecutive digits 0110 are found in two places, and three pairs of ones surrounded by zeroes can not occur in a sequence of 7 or 8 digits.) It is known that $v_n \lt v_{n+1} \lt v_{n+2}$ for some positive integer $n$. Prove that all the digits in the sequence $s$ are identical.