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Let S = {1, 2, . . . , 2021}, and let F denote the set of functions f : S → S. For a function f ∈ F, let Tf =  f 2021(s) : s ∈ S , where f 2021(s) denotes f(f(· · ·(f(s))· · ·)) with 2021 copies of f. Compute the remainder when sigma notation f∈F |Tf | is divided by the prime 2017, where the sum is over all functions f in F.

 Apr 8, 2024
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I guess you are talking about:

 

\(T_f = \{f^{2021}(s): s \in S\}\), find the remainder when \(\displaystyle \sum_{f\in F} |T_f|\) is divided by 2017, am I correct?

 

If I am correct, this is from HMMT 2021 Combinatorics Round, and the solution is here: https://hmmt-archive.s3.amazonaws.com/tournaments/2021/feb/comb/solutions.pdf

 Apr 9, 2024
edited by MaxWong  Apr 9, 2024

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