Question 137
Let $k$ be a positive integer. Find the maximal length $L$ of a sequence $a_1, \dots, a_L$ containing positive integers that satisfies the following two properties:
* Each term of the sequence is smaller or equal to $2^k$.
* There are no consecutive terms $a_i, a_{i+1}, \dots, a_j$ (with $1 \le i \lt j \le L$) and signs $s_i, s_{i+1}, \dots, s_j \in \{1, -1\}$ such that
$$s_i a_i + s_{i+1} a_{i+1} + \dots + s_j a_j = 0.$$
* Each term of the sequence is smaller or equal to $2^k$.
* There are no consecutive terms $a_i, a_{i+1}, \dots, a_j$ (with $1 \le i \lt j \le L$) and signs $s_i, s_{i+1}, \dots, s_j \in \{1, -1\}$ such that
$$s_i a_i + s_{i+1} a_{i+1} + \dots + s_j a_j = 0.$$