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 2 of 4: Use adjacency in a pentagon to get a contradiction
Detailed analysis
Among any three vertices of a pentagon, two are adjacent. The triangles through the chosen upper edge force the three corresponding edges from its other endpoint to have the opposite color. The adjacent pair among the three Bi then makes a monochromatic triangle, whichever color their base edge has. Thus every two upper edges incident at one vertex have the same color, so all five upper edges are one color.