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Contests
National and Regional Contests
Japan Contests
Japan MO Finals
2023 Japan MO Finals
5
5
Part of
2023 Japan MO Finals
Problems
(1)
Hard problem
Source: 2023 Japan MO Finals 5
2/11/2023
Let
S
=
{
1
,
2
,
…
,
3000
}
S=\{1,2,\dots,3000\}
S
=
{
1
,
2
,
…
,
3000
}
. Determine the maximum possible integer
X
X
X
that satisfies the condition: For all bijective function
f
:
S
→
S
f:S\rightarrow S
f
:
S
→
S
, there exists bijective function
g
:
S
→
S
g:S\rightarrow S
g
:
S
→
S
such that
∑
k
=
1
3000
(
max
{
f
(
f
(
k
)
)
,
f
(
g
(
k
)
)
,
g
(
f
(
k
)
)
,
g
(
g
(
k
)
)
}
−
min
{
f
(
f
(
k
)
)
,
f
(
g
(
k
)
)
,
g
(
f
(
k
)
)
,
g
(
g
(
k
)
)
}
)
≥
X
\displaystyle\sum_{k=1}^{3000}\left(\max\{f(f(k)),f(g(k)),g(f(k)),g(g(k))\}-\min\{f(f(k)),f(g(k)),g(f(k)),g(g(k))\}\right)\geq X
k
=
1
∑
3000
(
max
{
f
(
f
(
k
))
,
f
(
g
(
k
))
,
g
(
f
(
k
))
,
g
(
g
(
k
))}
−
min
{
f
(
f
(
k
))
,
f
(
g
(
k
))
,
g
(
f
(
k
))
,
g
(
g
(
k
))}
)
≥
X
combinatorics