1. (30th Russian Mathematical Olympiad) The positive integers from 1 to 100 are arranged in a circle in such a way that each number is either greater than both of its neighbors or less than both of its neighbors. A pair of adjacent numbers is called "good" if, after deleting them, the above property still holds. What is the minimum possible number of "good" pairs of neighbors?
1. The minimum possible number of "good" neighboring pairs is 51.

For example, first arrange the numbers from 1 to 100 in a clockwise direction in increasing order on a circle. Then, swap the positions of 2 and 3, 4 and 5, ..., 98 and 99. In the resulting arrangement \(1, 3, 2, 5, 4, \cdots, 99, 98, 100\), there are exactly 51 pairs of "good" neighbors.
$$
(1,3),(3,2),(5,4),(7,6), \cdots,(97,96),(99,98),(98,100) \text {. }
$$

Next, we prove that the number of "good" neighboring pairs is at least 51. To do this, we first prove that in any 2 intersecting "pairs," at least one pair is good.
Assume there are 5 consecutively placed numbers \(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\) such that \((a_{2}, a_{3})\) and \((a_{3}, a_{4})\) are both not good.
Without loss of generality, we can assume \(a_{1} > a_{2} < a_{3} > a_{4} < a_{5}\). On the other hand, since the pair \((a_{3}, a_{4})\) is not good, it must be that \(a_{1} > a_{2} > a_{5}\). Also, since the pair \((a_{2}, a_{3})\) is not good, it must be that \(a_{1} < a_{2} < a_{5}\). This leads to a contradiction: \(a_{1} > a_{2} > a_{5}\) and \(a_{1} < a_{2} < a_{5}\) cannot both be true.

Thus, there are at least 50 "good" neighboring pairs. If there are exactly 50 pairs, then "good" and "not good" neighboring pairs must alternate. However, if we consider the numbers 100 and its two neighbors \(a_{i-2}, a_{i-1}, 100, a_{i}, a_{i+1}\), it is easy to see that the pair \((a_{i}, a_{i+1})\) must be good, because \(100 > a_{i} < a_{i+1}\) or \(100 < a_{i} > a_{i+1}\). Similarly, the pair \((a_{i-2}, a_{i-1})\) must also be good. This indicates that "good" and "not good" neighboring pairs cannot alternate.