Question 80
Call a tuple $(a_1, \dots, a_n)$ of real numbers *stable* if the sums $a_1 + a_2 + \dots + a_k$, as well as the sums $a_k + a_{k+1} + \dots + a_n$, where in both cases $0 \lt k \le n$, are either all negative or all non-negative.
For instance, the tuple $(3, -1, 2)$ is stable, since:
$$
\begin{array}{lclcl}
3 & \ge & 0, & 2 & \ge 0, \\
3 + (-1) & \ge & 0, & (-1) + 2 & \ge 0, \\
3 + (-1) + 2 & \ge & 0; & 3 + (-1) + 2 & \ge 0.
\end{array}
$$
Prove that in any stable tuple with at least 3 terms where all terms are alternately negative and non-negative (it is unknown whether the first term is negative or non-negative), there exist 3 consecutive terms that together (without reordering) form a stable tuple on their own.
For instance, the tuple $(3, -1, 2)$ is stable, since:
$$
\begin{array}{lclcl}
3 & \ge & 0, & 2 & \ge 0, \\
3 + (-1) & \ge & 0, & (-1) + 2 & \ge 0, \\
3 + (-1) + 2 & \ge & 0; & 3 + (-1) + 2 & \ge 0.
\end{array}
$$
Prove that in any stable tuple with at least 3 terms where all terms are alternately negative and non-negative (it is unknown whether the first term is negative or non-negative), there exist 3 consecutive terms that together (without reordering) form a stable tuple on their own.