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Problems
Contests
National and Regional Contests
Japan Contests
Japan MO Finals
1997 Japan MO Finals
1997 Japan MO Finals
Part of
Japan MO Finals
Subcontests
(5)
5
1
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Proof of existing assignment for a line of letters
The letters
A
A
A
or
B
B
B
are assigned on the points divided equally into
2
n
(
n
=
1
,
2
,
⋯
)
2^{n}\ (n=1,\ 2,\cdots)
2
n
(
n
=
1
,
2
,
⋯
)
parts of a circumference.If you choose
n
n
n
letters from any succesively arranging points directed clockwise, prove that there exists the way of assignning for which the line of letters are mutually distinct.
4
1
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Condition of minimizing ax+bx+cx+dx
Let
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
be points in space which are not on a same plane and no any 3 points are not colinear. Suppose that the sum of the segments
A
X
+
B
X
+
C
X
+
D
X
AX+BX+CX+DX
A
X
+
BX
+
CX
+
D
X
is minimized at
X
=
X
0
X=X_0
X
=
X
0
which is different from
A
,
B
,
C
,
D
.
A,B,C,D.
A
,
B
,
C
,
D
.
Prove that
∠
A
X
0
B
=
∠
C
X
0
D
.
\angle{AX_0B}=\angle{CX_0D}.
∠
A
X
0
B
=
∠
C
X
0
D
.
3
1
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Minimum possible number of edges in the graph
Call the Graph the set which composed of several vertices
P
1
,
⋯
P
2
P_1,\ \cdots P_2
P
1
,
⋯
P
2
and several edges
(
(
(
segments
)
)
)
connecting two points among these vertices. Now let
G
G
G
be a graph with 9 vertices and satisfies the following condition.Condition: Even if we select any five points from the vertices in
G
,
G,
G
,
there exist at least two edges whose endpoints are included in the set of 5 points.What is the minimum possible numbers of edges satisfying the condition?
1
1
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10 points and circle
Take 10 points inside the circle with diameter 5. Prove that for any these 10 points there exist two points whose distance is less than 2.
2
1
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Japan 1997 inequality
Prove that
(
b
+
c
−
a
)
2
(
b
+
c
)
2
+
a
2
+
(
c
+
a
−
b
)
2
(
c
+
a
)
2
+
b
2
+
(
a
+
b
−
c
)
2
(
a
+
b
)
2
+
c
2
≥
3
5
\frac{\left(b+c-a\right)^{2}}{\left(b+c\right)^{2}+a^{2}}+\frac{\left(c+a-b\right)^{2}}{\left(c+a\right)^{2}+b^{2}}+\frac{\left(a+b-c\right)^{2}}{\left(a+b\right)^{2}+c^{2}}\geq\frac35
(
b
+
c
)
2
+
a
2
(
b
+
c
−
a
)
2
+
(
c
+
a
)
2
+
b
2
(
c
+
a
−
b
)
2
+
(
a
+
b
)
2
+
c
2
(
a
+
b
−
c
)
2
≥
5
3
for any positive real numbers
a
a
a
,
b
b
b
,
c
c
c
.