Question 70
Every term of the sequence $a_1, a_2, a_3, \dots$ is either $0$ or $1$. It is known that both $0$ and $1$ occur at least $1010$ times among every $2021$ consecutive terms of the sequence. May one be sure that the sequence is periodic from some place on, i.e., there exist positive integers $n$ and $p$ such that $a_{n+i} = a_{n+i+p}$ for every natural number $i$?