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Problems
Contests
National and Regional Contests
Japan Contests
Japan MO Finals
1992 Japan MO Finals
1992 Japan MO Finals
Part of
Japan MO Finals
Subcontests
(5)
5
1
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Dividing into Two pair with Sum n
Suppose that
n
≥
2
n\geq 2
n
≥
2
be integer and
a
1
,
a
2
,
a
3
,
a
4
a_1,\ a_2,\ a_3,\ a_4
a
1
,
a
2
,
a
3
,
a
4
satisfy the following condition: i)
n
\ n
n
and a_i\ (i \equal{} 1,\ 2,\ 3,\ 4) are relatively prime. ii) \ (ka_1)_n \plus{} (ka_2)_n \plus{} (ka_3)_n \plus{} (ka_4)_n \equal{} 2n for k \equal{} 1,\ 2,\ \cdots ,\ n \minus{} 1. Note that
(
a
)
n
(a)_n
(
a
)
n
expresses the divisor when
a
a
a
is divided by
n
n
n
. Prove that
(
a
1
)
n
,
(
a
2
)
n
,
(
a
3
)
n
,
(
a
4
)
n
(a_1)_n,\ (a_2)_n,\ (a_3)_n,\ (a_4)_n
(
a
1
)
n
,
(
a
2
)
n
,
(
a
3
)
n
,
(
a
4
)
n
can be divided into two pair with sum
n
n
n
.
4
1
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Mapping and Number of element of set
Let
A
A
A
be a
m
×
n
(
m
≠
n
)
m\times n\ (m\neq n)
m
×
n
(
m
=
n
)
matrix with the entries
0
0
0
and
1
1
1
. Suppose that if
f
f
f
is injective such that
f
:
{
1
,
2
,
⋯
,
m
}
⟶
{
1
,
2
,
⋯
,
n
}
f: \{1,\ 2,\ \cdots ,\ m\}\longrightarrow \{1,\ 2,\ \cdots ,\ n\}
f
:
{
1
,
2
,
⋯
,
m
}
⟶
{
1
,
2
,
⋯
,
n
}
, then there exists
1
≤
i
≤
m
1\leq i\leq m
1
≤
i
≤
m
such that
(
i
,
f
(
i
)
)
(i,\ f(i))
(
i
,
f
(
i
))
element is
0
0
0
. Prove that there exist
S
⊆
{
1
,
2
,
⋯
,
m
}
S\subseteq \{1,\ 2,\ \cdots ,\ m\}
S
⊆
{
1
,
2
,
⋯
,
m
}
and
T
⊆
{
1
,
2
,
⋯
,
n
}
T\subseteq \{1,\ 2,\ \cdots ,\ n\}
T
⊆
{
1
,
2
,
⋯
,
n
}
satisfying the condition:
i
)
i)
i
)
if
i
∈
S
,
j
∈
T
i\in{S},\ j\in{T}
i
∈
S
,
j
∈
T
, then
(
i
,
j
)
(i,\ j)
(
i
,
j
)
entry is
0
0
0
.
i
i
)
ii)
ii
)
card\ S \plus{} card\ T > n.
3
1
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Sigma and Inequality
Prove the inequality \sum_{k\equal{}1}^{n\minus{}1} \frac{n}{n\minus{}k}\cdot \frac{1}{2^{k\minus{}1}}<4\ (n\geq 2).
2
1
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Maximum Area of PDE
Suppose that
D
,
E
D,\ E
D
,
E
are points on
A
B
,
A
C
AB,\ AC
A
B
,
A
C
of
△
A
B
C
\triangle{ABC}
△
A
BC
with area
1
1
1
respectively and
P
P
P
is
B
E
∩
C
D
BE\cap CD
BE
∩
C
D
. When
D
,
E
D,\ E
D
,
E
move on
A
B
,
A
C
AB,\ AC
A
B
,
A
C
with satisfying the condition [BCED] \equal{} 2\triangle{PBC}, find the maximum area of
△
P
D
E
\triangle{PDE}
△
P
D
E
.
1
1
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x+y isn't divisor of x^n+y^n
Let
x
,
y
x,\ y
x
,
y
be relatively prime numbers such that
x
y
≠
1
xy\neq 1
x
y
=
1
. For positive even integer
n
n
n
, prove that x \plus{} y isn't a divisor of x^n \plus{} y^n.