Problem 5

Find a sequence of maximal length consisting of non-zero integers in which the sum of any seven consecutive terms is positive and that of any eleven consecutive terms is negative.

Step 3 of 5: Sum the same array by columns

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S=\sum_{j=0}^{10}\sum_{i=1}^{7}a_{i+j}>0\quad\text{(by columns)}

Detailed analysis

Every column is the sum of seven consecutive terms, so the hypothesis makes every column sum positive. The same total $S$ must therefore be positive, contradicting the preceding step. Hence no valid sequence has length greater than $16$ .