Constructive Counting
AMC 8 LEVEL PROBLEMS
Refer to this page for a good overview of constructive counting technique.
AIME LEVEL PROBLEMS
How to properly construct a scenario to reduce the complexity of a problem is the most challenging aspect of this type of problem. The task at hand usually seems daunting at first look -- large number, or a complicated setup, etc. -- it often helps to start simple, use a small number to find the pattern, then go from there.
Considering how challenging these problems can be, I offered some hints at the bottom of the page. Try to solve the problems. If you are stuck, refer to the hints and see if it helps you construct a simpler model to approach the problem.
2023 AIME II Problem 7 Problem 10
HINTS:
2023 AIME II Problem 7: The key to solving this problem is to realize that there can only be at most one same-colored pair
2023 AIME II Problem 10: The key here is to represent 12 numbers as x mod(3), effectively reducing them to 3 groups of 4, then to realize that each column can only have numbers from two different groups.