2022 USAJMO Problems/Problem 3
Problem
Let and be fixed integers, and . Given are identical black rods and identical white rods, each of side length 1.
We assemble a regular -gon using these rods so that parallel sides are the same color. Then, a convex -gon is formed by translating the black rods, and a convex -gon is formed by translating the white rods. An example of one way of doing the assembly when and is shown below, as well as the resulting polygons and .
[image here]
Prove that the difference of the areas of and depends only on the numbers and , and not on how the -gon was assembled.