[Solution] By cutting along the dotted lines in the 1000-sided polygon shown in the figure, we can obtain 501 triangles.

Note, in each of the triangles obtained by cutting, one side is the cut, and the other two sides are cut from adjacent sides of the original polygon. Moreover, all triangles on the same side of the cut, except for their cut side, have edges from different sides of the original polygon; triangles on opposite sides of the cut can only have one side from the same original side of the polygon if they are the leftmost or rightmost triangles. Therefore, the polygon can be cut into at most 501 triangles.
In summary, a 1000-sided polygon can be cut into at most 501 triangles.
$7 \cdot 30$ Cutting a 1000-sided polygon (not necessarily convex) made of thick cardboard along a certain straight line once, it divides it into several new polygons. How many triangles can there be at most among them?