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2020 China Mathematics Olympiad

Source: 2020 CMO P3

November 26, 2019
combinatorics

Problem Statement

Let SS be a set, ∣S∣=35|S|=35. A set FF of mappings from SS to itself is called to be satisfying property P(k)P(k), if for any x,y∈Sx,y\in S, there exist f1,⋯ ,fk∈Ff_1, \cdots, f_k \in F (not necessarily different), such that fk(fk−1(⋯(f1(x))))=fk(fk−1(⋯(f1(y))))f_k(f_{k-1}(\cdots (f_1(x))))=f_k(f_{k-1}(\cdots (f_1(y)))). Find the least positive integer mm, such that if FF satisfies property P(2019)P(2019), then it also satisfies property P(m)P(m).
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