RU-ALLRUSSIAN 1982 Problem 18

Given a real sequence $\mathbf{a}_{1}, \mathbf{a}_{2}, \ldots , \mathbf{a}_{\mathrm{n}}$ , show that it is always possible to choose a subsequence such that (1) for each $\mathrm{i} \leq \mathrm{n} - 2$ at least one and at most two of $\mathbf{a}_{\mathrm{i}}, \mathbf{a}_{\mathrm{i} + 1}, \mathbf{a}_{\mathrm{i} + 2}$ are chosen and (2) the sum of the absolute values of the numbers in the subsequence is at least $1 / 6 \sum \mathrm{n}^{\mathrm{i}} \mathrm{la}_{\mathrm{i}}$ .