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China Contests
China National Olympiad
2020 China National Olympiad
3
3
Part of
2020 China National Olympiad
Problems
(1)
2020 China Mathematics Olympiad
Source: 2020 CMO P3
11/26/2019
Let
S
S
S
be a set,
∣
S
∣
=
35
|S|=35
∣
S
∣
=
35
. A set
F
F
F
of mappings from
S
S
S
to itself is called to be satisfying property
P
(
k
)
P(k)
P
(
k
)
, if for any
x
,
y
∈
S
x,y\in S
x
,
y
∈
S
, there exist
f
1
,
⋯
,
f
k
∈
F
f_1, \cdots, f_k \in F
f
1
,
⋯
,
f
k
∈
F
(not necessarily different), such that
f
k
(
f
k
−
1
(
⋯
(
f
1
(
x
)
)
)
)
=
f
k
(
f
k
−
1
(
⋯
(
f
1
(
y
)
)
)
)
f_k(f_{k-1}(\cdots (f_1(x))))=f_k(f_{k-1}(\cdots (f_1(y))))
f
k
(
f
k
−
1
(
⋯
(
f
1
(
x
))))
=
f
k
(
f
k
−
1
(
⋯
(
f
1
(
y
))))
. Find the least positive integer
m
m
m
, such that if
F
F
F
satisfies property
P
(
2019
)
P(2019)
P
(
2019
)
, then it also satisfies property
P
(
m
)
P(m)
P
(
m
)
.
combinatorics