Question 64
Determine the maximum positive integer $L$ such that there exists a sequence $a_1, a_2, \dots, a_L$ of positive integers satisfying:
(a) every term in the sequence is less than or equal to $2^{2024}$;
(b) there does NOT exist a consecutive subsequence $a_i, a_{i+1}, \dots, a_j$ (where $1 \le i \le j \le L$) with a choice of signs $s_i, s_{i+1}, \dots, s_j \in \{-1, 1\}$ for which
$$
s_i a_i + s_{i+1} a_{i+1} + \dots + s_j a_j = 0.
$$
(a) every term in the sequence is less than or equal to $2^{2024}$;
(b) there does NOT exist a consecutive subsequence $a_i, a_{i+1}, \dots, a_j$ (where $1 \le i \le j \le L$) with a choice of signs $s_i, s_{i+1}, \dots, s_j \in \{-1, 1\}$ for which
$$
s_i a_i + s_{i+1} a_{i+1} + \dots + s_j a_j = 0.
$$