8. 8.1. The vertices of a regular 22-sided polygon are numbered. In how many ways can four of its vertices be chosen to form a trapezoid? (A trapezoid is a quadrilateral with one pair of parallel sides and the other two sides not parallel).
Answer: 990.

## Solution.

Note that all trapezoids we obtain will be isosceles. We will count the number of inscribed quadrilaterals that have an axis of symmetry that does not pass through the vertices. Thus, we will count all trapezoids once (since each isosceles trapezoid has only one axis of symmetry), and count each rectangle twice (since a rectangle has two such axes of symmetry).

The axis of symmetry we consider can contain vertices of the polygon or not, see the figure.
![](https://cdn.mathpix.com/cropped/2024_05_06_6a3c59b5c78061150e51g-6.jpg?height=500&width=1482&top_left_y=590&top_left_x=360)

Now let's count the number of the described quadrilaterals in a $2n$-gon. 1) Consider an axis of symmetry passing through the midpoint of a side of the $2n$-gon. There are exactly $n$ such axes. Choosing any two points on one side of the axis of symmetry uniquely determines our quadrilateral. The number of ways to choose two points from $n$ is $C_{n}^{2}=n(n-1) / 2$. Thus, in this case, we have counted $n^{2}(n-1) / 2$ quadrilaterals. 2) If the axis of symmetry passes through two vertices of the polygon, there are again $n$ such axes. Choosing any two points on one side of the axis of symmetry also uniquely determines our quadrilateral, but now there are $n-1$ points on each side of the axis. The number of ways to choose two points from $n-1$ is $C_{n-1}^{2}=(n-1)(n-2) / 2$. Thus, in this case, we have $n(n-1)(n-2) / 2$ quadrilaterals. We need to subtract twice the number of rectangles, and we will get the answer. Each rectangle is uniquely determined by two diameters of the circle. Therefore, their doubled number is $2 C_{n}^{2}=n(n-1)$. Thus, we finally get $n^{2}(n-1) / 2+n(n-1)(n-2) / 2-2 n(n-1) / 2=n(n-1)(n-2)$. In the case of a 22-gon, $n=11$ and the number of ways to choose a trapezoid will be $11 \cdot 10 \cdot 9=990$.