8. (4 points) A triangle is divided into 1000 triangles. What is the maximum number of different points at which the vertices of these triangles can be located?
Answer: 1002

Solution: The sum of the angles of a triangle is $180^{\circ}$, thousands of triangles $-180000^{\circ}$. Where did the extra $179820^{\circ}$ come from? Each internal vertex, where only the angles of triangles meet, adds $360^{\circ}$. Each vertex on the side of the original triangle or on the side of one of the triangles in the partition adds $180^{\circ}$. To achieve the maximum, it is necessary that there are only points of the second type and there are $179820: 180=999$ of them. An example is constructed by sequentially adding new vertices to the sides of the existing triangles.

Adding the three vertices of the original triangle, we get the answer 1002.

## 7th grade

2nd variant