We are considering all possible triangles with integer sides and a perimeter of 2000, as well as all possible triangles with integer sides and a perimeter of 2003. Which set of triangles is larger?
## Solution

Let the sides of the triangle be integers $a, b, c$ and its perimeter be 2000. We will associate this triangle with a triangle with sides $a+1, b+1, c+1$, whose perimeter is 2003 (it is easy to verify that the triangle inequality holds for these sides). Under this correspondence, different triangles correspond to different ones. However, the triangle with a perimeter of 2003 and sides 1, 1001, 1001 does not correspond to any triangle with a perimeter of 2000. Therefore, there are more triangles with a perimeter of 2003.

## Answer

With a perimeter of 2003.