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
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
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