A thousand points are the vertices of a convex thousand-sided polygon, inside which there are another five hundred points such that no three of the five hundred lie on the same line. This thousand-sided polygon is cut into triangles in such a way that all the specified 1500 points are vertices of the triangles and these triangles have no other vertices. How many triangles will result from such a cutting?
Answer: 1998. The sum of all angles of the obtained triangles is equal to the sum of the angles of a 1000-gon and 500 angles of $360^{\circ}$, corresponding to 500 internal points. Therefore, the sum of the angles of the triangles is $998 \cdot 180^{\circ} + 500 \cdot 360^{\circ}$ $=(998 + 2 \cdot 500) 180^{\circ}$. Hence, the number of obtained triangles is $998 + 2 \cdot 500 = 1998$.