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Problems
Contests
National and Regional Contests
Japan Contests
Japan MO Finals
2001 Japan MO Finals
2001 Japan MO Finals
Part of
Japan MO Finals
Subcontests
(5)
2
1
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n is the product of one more than each digit
An integer
n
>
0
n>0
n
>
0
is written in decimal system as
a
m
a
m
−
1
…
a
1
‾
\overline{a_ma_{m-1}\ldots a_1}
a
m
a
m
−
1
…
a
1
. Find all
n
n
n
such that
n
=
(
a
m
+
1
)
(
a
m
−
1
+
1
)
⋯
(
a
1
+
1
)
n=(a_m+1)(a_{m-1}+1)\cdots (a_1+1)
n
=
(
a
m
+
1
)
(
a
m
−
1
+
1
)
⋯
(
a
1
+
1
)
4
1
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Japan-2001(4)
Let
p
p
p
be a prime number and
m
m
m
a positive integer. Show that there exists a positive integer
n
n
n
such that the decimal representation of
p
n
p^n
p
n
contains a string of
m
m
m
consecutive zeros.
5
1
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A hard problem (From Japan 2001)
Suppose that
A
B
C
ABC
A
BC
and
P
Q
R
PQR
PQR
are triangles such that
A
,
P
A,P
A
,
P
are the midpoints of
Q
R
,
B
C
QR,BC
QR
,
BC
respectively, and
Q
R
,
B
C
QR,BC
QR
,
BC
are the bisectors of
∠
B
A
C
,
∠
Q
P
R
\angle BAC,\angle QPR
∠
B
A
C
,
∠
QPR
. Prove that
A
B
+
A
C
=
P
Q
+
P
R
AB+AC=PQ+PR
A
B
+
A
C
=
PQ
+
PR
.
1
1
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JAPANESE 2001
Each square of an
m
×
n
m\times n
m
×
n
chessboard is painted black or white in such a way that for every black square, the number of black squares adjacent to it is odd (two squares are adjacent if they share one edge). Prove that the number of black squares is even.
3
1
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japan 2001
Three nonnegative real numbers satisfy
a
,
b
,
c
a,b,c
a
,
b
,
c
satisfy
a
2
≤
b
2
+
c
2
,
b
2
≤
c
2
+
a
2
a^2\le b^2+c^2, b^2\le c^2+a^2
a
2
≤
b
2
+
c
2
,
b
2
≤
c
2
+
a
2
and
c
2
≤
a
2
+
b
2
c^2\le a^2+b^2
c
2
≤
a
2
+
b
2
. Prove the inequality
(
a
+
b
+
c
)
(
a
2
+
b
2
+
c
2
)
(
a
3
+
b
3
+
c
3
)
≥
4
(
a
6
+
b
6
+
c
6
)
.
(a+b+c)(a^2+b^2+c^2)(a^3+b^3+c^3)\ge 4(a^6+b^6+c^6).
(
a
+
b
+
c
)
(
a
2
+
b
2
+
c
2
)
(
a
3
+
b
3
+
c
3
)
≥
4
(
a
6
+
b
6
+
c
6
)
.
When does equality hold?