1. The midpoints of adjacent sides of a rectangle with a perimeter of 32 were connected by segments. The same operation was performed on the resulting quadrilateral: the midpoints of adjacent sides were connected by segments (see figure). How many times in total does one need to perform such an operation so that the perimeter of the resulting quadrilateral first becomes less than 1?
Answer: 11.

Solution: After two operations, a quadrilateral is obtained, the sides of which are the midlines of triangles with bases parallel to the sides of the original rectangle. Therefore, this quadrilateral is a rectangle, and each of its sides is half the length of the corresponding side of the original rectangle. Thus, after 2 operations, the perimeter is halved. After 5 pairs of operations, the perimeter will decrease by a factor of $2^{5}=32$ and will become equal to 1. From the triangle inequality, it is clear that the perimeter decreases with each operation. Therefore, after 11 operations, the perimeter will be less than 1.

![](https://cdn.mathpix.com/cropped/2024_05_06_8687bd1ddeb4733a341eg-1.jpg?height=229&width=368&top_left_y=1413&top_left_x=1598)

Comment: A specific case is considered (for example, calculations are performed for a rectangle with sides given by specific numbers), but the method of calculations can be applied to any rectangle - 4 points; the method is applicable only to rectangles of a certain type - 1-2 points. If the first two operations were not taken into account when counting the number of operations, deduct 1 point. If it is not proven that the perimeter decreases with each operation, deduct 1 point.