National Olympiad Final Round

Question

Call a tuple (a_1, , a_n) of real numbers *stable* if the sums a_1 + a_2 + + a_k, as well as the sums a_k + a_k+1 + + a_n, where in both cases 0 < k n, are either all negative or all non-negative. For instance, the tuple (3, -1, 2) is stable, since: arraylclcl 3 & & 0, & 2 & 0, 3 + (-1) & & 0, & (-1) + 2 & 0, 3 + (-1) + 2 & & 0; & 3 + (-1) + 2 & 0. array Prove that in any stable tuple with at least 3 terms where all terms are alternately negative and non-negative (it is unknown whether the first term is negative or non-negative), there exist 3 consecutive terms that together (without reordering) form a stable tuple on their own.

Accepted Answer

Consider terms whose absolute value is minimal in the tuple. If there exists a negative such element, denote it a_i, then the sum of a_i and its any neighbour is non-negative. Thus a_i is neither the first nor the last in the tuple because of stability of the tuple. But then both a_i-1 + a_i and a_i + a_i+1 are non-negative, as well as a_i-1 + a_i + a_i+1, hence a_i-1, a_i, a_i+1 together form a stable subtuple. If all elements with minimal absolute value are non-negative then let a_i be any of them. Analogously to the previous case, both a_i-1 + a_i and a_i + a_i+1 are negative, as well as a_i-1 + a_i + a_i+1, hence a_i-1, a_i, a_i+1 together form a stable tuple.