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Problems
Contests
National and Regional Contests
China Contests
China Team Selection Test
2003 China Team Selection Test
1
Find number of elements
Find number of elements
Source: China TST 2003
June 29, 2006
algebra unsolved
algebra
Problem Statement
m
m
m
and
n
n
n
are positive integers. Set
A
=
{
1
,
2
,
⋯
,
n
}
A=\{ 1, 2, \cdots, n \}
A
=
{
1
,
2
,
⋯
,
n
}
. Let set
B
n
m
=
{
(
a
1
,
a
2
⋯
,
a
m
)
∣
a
i
∈
A
,
i
=
1
,
2
,
⋯
,
m
}
B_{n}^{m}=\{ (a_1, a_2 \cdots, a_m) \mid a_i \in A, i= 1, 2, \cdots, m \}
B
n
m
=
{(
a
1
,
a
2
⋯
,
a
m
)
∣
a
i
∈
A
,
i
=
1
,
2
,
⋯
,
m
}
satisfying: (1)
∣
a
i
−
a
i
+
1
∣
≠
n
−
1
|a_i - a_{i+1}| \neq n-1
∣
a
i
−
a
i
+
1
∣
=
n
−
1
,
i
=
1
,
2
,
⋯
,
m
−
1
i=1,2, \cdots, m-1
i
=
1
,
2
,
⋯
,
m
−
1
; and (2) at least three of
a
1
,
a
2
,
⋯
,
a
m
a_1, a_2, \cdots, a_m
a
1
,
a
2
,
⋯
,
a
m
(
m
≥
3
m \geq 3
m
≥
3
) are pairwise distince. Find
∣
B
n
m
∣
|B_n^m|
∣
B
n
m
∣
and
∣
B
6
3
∣
|B_6^3|
∣
B
6
3
∣
.
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