12. Let $x_{1} \ldots x_{n}$ 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 terms is always strictly negative.

Which of the following statements is true?

(A) The sequence has at most 18 terms

(B) The sequence can have 19 terms

(C) The sequence has exactly 21 terms

(D) The sequence has at least 18 terms

(E) There exist sequences of arbitrary length that satisfy i) and ii).

## Numerical answer problems -5 points
12. The answer is (A). It is enough to observe that if the sequence has more than 18 elements, then the quantity:

$$
X=\left(x_{1}+\ldots+x_{7}\right)+\left(x_{2}+\ldots+x_{8}\right)+\ldots+\left(x_{13}+\ldots+x_{19}\right)
$$

is both strictly greater than zero and strictly less than zero. Indeed: it is strictly greater than zero due to condition (i), while it is less than zero because, provided we rearrange the terms of the right-hand sum, we have that:

$$
X=\left(x_{1}+\ldots+x_{13}\right)+\left(x_{2}+\ldots+x_{14}\right)+\ldots+\left(x_{7}+\ldots+x_{19}\right)
$$

which, by (ii), is a quantity strictly less than zero. Therefore, the answer can only be (A).

(Alfieri)