Question 36
Call an $n$-tuple $(a_1, \dots, a_n)$ of real numbers stable if the sums $a_1 + a_2 + \dots + a_k$ where $0 \lt k \le n$, as well as the sums $a_n + a_{n-1} + \dots + a_{n-k}$ where $0 \le k \lt n$, are either all negative or all non-negative.
Let $k$ be any natural number. Consider all stable $(2k+1)$-tuples consisting of real numbers that are alternately negative and non-negative. Find the least possible number of stable subtuples with more than one element that can be contained in such a tuple.
(A Subtuple of $(a_1, \dots, a_n)$ is any tuple $(a_i, \dots, a_j)$, $1 \le i \le j \le n$, of elements consecutive in the original tuple.)
Let $k$ be any natural number. Consider all stable $(2k+1)$-tuples consisting of real numbers that are alternately negative and non-negative. Find the least possible number of stable subtuples with more than one element that can be contained in such a tuple.
(A Subtuple of $(a_1, \dots, a_n)$ is any tuple $(a_i, \dots, a_j)$, $1 \le i \le j \le n$, of elements consecutive in the original tuple.)