2. Given a regular 22-gon $M$. Find the number of quadruples of vertices of this 22-gon that are the vertices of convex quadrilaterals, which have at least one pair of parallel sides.
Answer: 1045.

Solution. Let's inscribe the given polygon $K_{1} K_{2} \ldots K_{22}$ in a circle. Each quadrilateral with a pair of parallel sides is determined by a pair of parallel chords with endpoints at points $K_{1}, \ldots, K_{22}$.

Consider a chord connecting two adjacent vertices of the polygon, for example, $K_{6} K_{7}$. There are 10 more chords parallel to it ( $K_{5} K_{8}$, etc.), i.e., we get a set of 11 chords parallel to each other. From these, we can form $C_{11}^{2}=55$ pairs of parallel segments.

We will get similar sets if we consider all chords parallel to $K_{1} K_{2}, \ldots, K_{11} K_{12}$ - a total of 11 such sets.

Now, let's take a chord connecting vertices that are one apart from each other, for example, $K_{6} K_{8}$. There are 9 more chords parallel to it ( $K_{5} K_{9}$, etc.), i.e., we get a set of 10 chords parallel to each other. From these, we can form $C_{10}^{2}=45$ pairs of parallel segments. There are also 11 such sets.

In the end, we get $11 \cdot 55 + 11 \cdot 45 = 1100$ quadrilaterals. With this method of counting, rectangles were counted twice. Note that both diagonals of an inscribed rectangle in a circle are diameters, and there are 11 diameters with vertices at the given points, thus we get $C_{11}^{2}=55$ rectangles. Therefore, we have $1100 - 55 = 1045$ quadrilaterals with at least one pair of parallel sides.