Problem:
Let be a finite sequence of real numbers such that:
i) the sum of any 7 consecutive terms is always strictly positive;
ii) the sum of any 13 consecutive elements is always strictly negative.
Which of the following statements is true?
Italian Mathematical Olympiad
Pick one
Solution
Solution:
The answer is (A). It suffices to observe that if the sequence has more than 18 elements then the quantity:
is both strictly greater than zero and strictly less than zero; indeed: it is strictly greater than zero by condition (i), while it is less than zero since, upon rearranging the terms on the right-hand side, we have that:
which by (ii) is a quantity strictly less than zero. Hence the answer can only be (A).
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