Lor
2023 AIME II
Problem 7
Each vertex of a regular dodecagon (
-gon) is to be colored either red or blue, and thus there are 
possible colorings. Find the number of these colorings with the property that no four vertices colored the same color are the four vertices of a rectangle.
Consider these diameters A1A7, A2A8, A3A9, A4A10, A5A11, A6A12
If there exist a rectangle with four vertices colored by same color then four endpoints of 2 of the diameters should have same color
Case1) 2 endpoints of all 6 diameters have different colors
Case2) 2 endpoints of only one diameter have same color
Case3) 2 endpoints of only 2 diameters have same color
Solution