Problem 2
A prism with pentagons and as top and bottom faces is given. Each side of the two pentagons and each segment is colored red or green. Every triangle whose vertices are vertices of the prism and whose sides have all been colored has two sides of different colors. Prove that all 10 sides of the top and bottom faces have the same color.
Step 4 of 4: Show the two monochromatic colors agree
Detailed analysis
Suppose the upper edges are green and the lower edges red. At A1, at least three cross-edges must be green: if three were red, two corresponding Bi would be adjacent and form a red triangle with A1. The same holds at A2, so two sets of three green cross-edges in a five-element set intersect. A common Bi would make the green triangle A1A2Bi, contradiction.