Here's a Math Olympiad question that has an idea similar to Green's theorem in its solution. Unfortunately, I didn't find the source other than my recollection.

The depicted planar graph has 24 triangles formed by its edges and 19 nodes. Each node is arbitrarily assigned a distinct integer; the assignment gives each edge orientation from the smaller integer to the larger one. Prove that at least 7 of the triangles are oriented clockwise.