Question 83
Assume that $S = (a_1, a_2, \dots, a_n)$ consists of $0$ and $1$ and is the longest sequence of numbers which satisfies the following condition: Every two sections of successive $5$ terms in the sequence of numbers $S$ are different, i.e., for arbitrary $1 \leq i \lt j \leq n-4$, $(a_i, a_{i+1}, a_{i+2}, a_{i+3}, a_{i+4})$ and $(a_j, a_{j+1}, a_{j+2}, a_{j+3}, a_{j+4})$ are different. Prove that the first four terms and the last four terms in the sequence are the same.