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National and Regional Contests
Japan Contests
Japan MO Finals
2011 Japan MO Finals
2011 Japan MO Finals
Part of
Japan MO Finals
Subcontests
(5)
3
1
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2011 Japan Mathematical Olympiad Finals Problem 3
Person
A
A
A
writes down non negative integers in each
N
N
N
grid running in a line horizontally. When
A
A
A
says one non negative integer, Person
B
B
B
replaces some number in
N
N
N
grid by the number that
A
A
A
said. Repeat this procedure, when these numbers are arranged in the order of monotone increasing in the wider sense, the procedure is over. Is it possible that
B
B
B
can finish in regard less of
A
A
A
?
5
1
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2011 Japan Mathematical Olympiad Finals Problem 5
Given 4 points on a plane. Suppose radii of 4 incircles of the triangles, which can be formed by any 3 points taken from the 4 points, are equal. Prove that all of the triangles are congruent.
4
1
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2011 Japan Mathematical Olympiad Finals Problem 4
Find all functions
f
:
R
→
R
f : \mathbb{R} \to\mathbb{R}
f
:
R
→
R
such that
f
(
f
(
x
)
−
f
(
y
)
)
=
f
(
f
(
x
)
)
−
2
x
2
f
(
y
)
+
f
(
y
2
)
f(f(x)-f(y))=f(f(x))-2x^2f(y)+f(y^2)
f
(
f
(
x
)
−
f
(
y
))
=
f
(
f
(
x
))
−
2
x
2
f
(
y
)
+
f
(
y
2
)
for all
x
,
y
∈
R
.
x,y \in \mathbb{R}.
x
,
y
∈
R
.
2
1
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2011 Japan Mathematical Olympiad Finals Problem 2
Find all of quintuple of positive integers
(
a
,
n
,
p
,
q
,
r
)
(a,n,p,q,r)
(
a
,
n
,
p
,
q
,
r
)
such that
a
n
−
1
=
(
a
p
−
1
)
(
a
q
−
1
)
(
a
r
−
1
)
a^n-1=(a^p-1)(a^q-1)(a^r-1)
a
n
−
1
=
(
a
p
−
1
)
(
a
q
−
1
)
(
a
r
−
1
)
.
1
1
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2011 Japan Mathematical Olympiad Finals Problem 1
Given an acute triangle
A
B
C
ABC
A
BC
with the midpoint
M
M
M
of
B
C
BC
BC
. Draw the perpendicular
H
P
HP
H
P
from the orthocenter
H
H
H
of
A
B
C
ABC
A
BC
to
A
M
AM
A
M
. Show that
A
M
⋅
P
M
=
B
M
2
AM\cdot PM=BM^2
A
M
⋅
PM
=
B
M
2
.