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National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2020 Korea Junior Math Olympiad
3
3
Part of
2020 Korea Junior Math Olympiad
Problems
(1)
The Number of pairs
Source: Korea Junior Mathematics Olympiad 2020 P3
11/22/2020
The permutation
σ
\sigma
σ
consisting of four words
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
has
f
A
B
(
σ
)
f_{AB}(\sigma)
f
A
B
(
σ
)
, the sum of the number of
B
B
B
placed rightside of every
A
A
A
. We can define
f
B
C
(
σ
)
f_{BC}(\sigma)
f
BC
(
σ
)
,
f
C
D
(
σ
)
f_{CD}(\sigma)
f
C
D
(
σ
)
,
f
D
A
(
σ
)
f_{DA}(\sigma)
f
D
A
(
σ
)
as the same way too. For example,
σ
=
A
C
B
D
B
A
C
D
C
B
A
D
\sigma=ACBDBACDCBAD
σ
=
A
CB
D
B
A
C
D
CB
A
D
,
f
A
B
(
σ
)
=
3
+
1
+
0
=
4
f_{AB}(\sigma)=3+1+0=4
f
A
B
(
σ
)
=
3
+
1
+
0
=
4
,
f
B
C
(
σ
)
=
4
f_{BC}(\sigma)=4
f
BC
(
σ
)
=
4
,
f
C
D
(
σ
)
=
6
f_{CD}(\sigma)=6
f
C
D
(
σ
)
=
6
,
f
D
A
(
σ
)
=
3
f_{DA}(\sigma)=3
f
D
A
(
σ
)
=
3
Find the maximal value of
f
A
B
(
σ
)
+
f
B
C
(
σ
)
+
f
C
D
(
σ
)
+
f
D
A
(
σ
)
f_{AB}(\sigma)+f_{BC}(\sigma)+f_{CD}(\sigma)+f_{DA}(\sigma)
f
A
B
(
σ
)
+
f
BC
(
σ
)
+
f
C
D
(
σ
)
+
f
D
A
(
σ
)
, when
σ
\sigma
σ
consists of
2020
2020
2020
letters for each
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
combinatorics
KJMO