4. On a circle, 40 red points and one blue point are marked. All possible polygons with vertices at the marked points are considered. Which type of polygons is more numerous, and by how many: those with a blue vertex, or those without it?
Answer: The number of polygons with a blue vertex is 780 more than the number of polygons without a blue vertex.

Solution. Let's call polygons with a blue vertex blue, and those without a blue vertex red. Take an arbitrary red polygon. Adding a blue vertex to it gives exactly one blue polygon. Thus, any blue polygon with at least four vertices can be obtained this way. Therefore, the difference between the number of blue polygons and the number of red polygons is equal to the number of blue triangles. The number of blue triangles is the same as the number of ways to choose two red points (two red vertices of a blue triangle), i.e., $\frac{40 \cdot 39}{2} = 780$.

Evaluation. 14 points for a correct solution.

## Tasks, answers, and evaluation criteria

7th grade

Variant 1 physics