Question 28
Given a real sequence $a_1, a_2, \ldots, a_n$, show that it is always possible to choose a subsequence such that
(1) for each $i \leq n - 2$ at least one and at most two of $a_i, a_{i+1}, a_{i+2}$ are chosen, and
(2) the sum of the absolute values of the numbers in the subsequence is at least $\frac{1}{6} \sum_{i=1}^n |a_i|$.
(1) for each $i \leq n - 2$ at least one and at most two of $a_i, a_{i+1}, a_{i+2}$ are chosen, and
(2) the sum of the absolute values of the numbers in the subsequence is at least $\frac{1}{6} \sum_{i=1}^n |a_i|$.