Question 29
Let $n \gt 1$ be an integer. Find all sequences $a_{1}, a_{2}, \ldots, a_{n^{2}+n}$ satisfying the following conditions:
(a) $a_{i} \in \{0,1\}$ for all $1 \leq i \leq n^{2}+n$;
(b) $a_{i+1}+a_{i+2}+\ldots+a_{i+n} \lt a_{i+n+1}+a_{i+n+2}+\ldots+a_{i+2n}$ for all $0 \leq i \leq n^{2}-n$.
(a) $a_{i} \in \{0,1\}$ for all $1 \leq i \leq n^{2}+n$;
(b) $a_{i+1}+a_{i+2}+\ldots+a_{i+n} \lt a_{i+n+1}+a_{i+n+2}+\ldots+a_{i+2n}$ for all $0 \leq i \leq n^{2}-n$.