AIME I 2026 (Problem 7)
Find the number of functions mapping the set onto such that for every ,
Prove that is a one to one function
By Toolkit 125 (Functional Graphs) and Hint 2,
functional graph of is union of disjoint directed cycles.
functional graph of is union of disjoint directed cycles.
If the length of a cycle is .
We reach again after applying to times,
We also know that
We reach again after applying to times,
We also know that
Use Casework: Start with the Most Limiting Case and base your casework on the length of largest cycle .
By Hint 4
By Hint 4
Case 1:
WLOG assume the top vertex is .
WLOG assume the top vertex is .
Case 2:
Case 3:
Case 4:
By Hints 6,7,8, and 9
396