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Problems
Contests
International Contests
APMO
2010 APMO
2010 APMO
Part of
APMO
Subcontests
(5)
5
1
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Asian Pacific Mathematical Olympiad 2010 Problem 5
Find all functions
f
f
f
from the set
R
\mathbb{R}
R
of real numbers into
R
\mathbb{R}
R
which satisfy for all
x
,
y
,
z
∈
R
x, y, z \in \mathbb{R}
x
,
y
,
z
∈
R
the identity
f
(
f
(
x
)
+
f
(
y
)
+
f
(
z
)
)
=
f
(
f
(
x
)
−
f
(
y
)
)
+
f
(
2
x
y
+
f
(
z
)
)
+
2
f
(
x
z
−
y
z
)
.
f(f(x)+f(y)+f(z))=f(f(x)-f(y))+f(2xy+f(z))+2f(xz-yz).
f
(
f
(
x
)
+
f
(
y
)
+
f
(
z
))
=
f
(
f
(
x
)
−
f
(
y
))
+
f
(
2
x
y
+
f
(
z
))
+
2
f
(
x
z
−
yz
)
.
4
1
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Asian Pacific Mathematical Olympiad 2010 Problem 4
Let
A
B
C
ABC
A
BC
be an acute angled triangle satisfying the conditions
A
B
>
B
C
AB>BC
A
B
>
BC
and
A
C
>
B
C
AC>BC
A
C
>
BC
. Denote by
O
O
O
and
H
H
H
the circumcentre and orthocentre, respectively, of the triangle
A
B
C
.
ABC.
A
BC
.
Suppose that the circumcircle of the triangle
A
H
C
AHC
A
H
C
intersects the line
A
B
AB
A
B
at
M
M
M
different from
A
A
A
, and the circumcircle of the triangle
A
H
B
AHB
A
H
B
intersects the line
A
C
AC
A
C
at
N
N
N
different from
A
.
A.
A
.
Prove that the circumcentre of the triangle
M
N
H
MNH
MN
H
lies on the line
O
H
OH
O
H
.
3
1
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Asian Pacific Mathematical Olympiad 2010 Problem 3
Let
n
n
n
be a positive integer.
n
n
n
people take part in a certain party. For any pair of the participants, either the two are acquainted with each other or they are not. What is the maximum possible number of the pairs for which the two are not acquainted but have a common acquaintance among the participants?
2
1
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Asian Pacific Mathematical Olympiad 2010 Problem 2
For a positive integer
k
,
k,
k
,
call an integer a
p
u
r
e
pure
p
u
re
k
−
t
h
k-th
k
−
t
h
p
o
w
e
r
power
p
o
w
er
if it can be represented as
m
k
m^k
m
k
for some integer
m
.
m.
m
.
Show that for every positive integer
n
,
n,
n
,
there exists
n
n
n
distinct positive integers such that their sum is a pure
2009
−
2009-
2009
−
th power and their product is a pure
2010
−
2010-
2010
−
th power.
1
1
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Asian Pacific Mathematical Olympiad 2010 Problem 1
Let
A
B
C
ABC
A
BC
be a triangle with
∠
B
A
C
≠
9
0
∘
.
\angle BAC \neq 90^{\circ}.
∠
B
A
C
=
9
0
∘
.
Let
O
O
O
be the circumcenter of the triangle
A
B
C
ABC
A
BC
and
Γ
\Gamma
Γ
be the circumcircle of the triangle
B
O
C
.
BOC.
BOC
.
Suppose that
Γ
\Gamma
Γ
intersects the line segment
A
B
AB
A
B
at
P
P
P
different from
B
B
B
, and the line segment
A
C
AC
A
C
at
Q
Q
Q
different from
C
.
C.
C
.
Let
O
N
ON
ON
be the diameter of the circle
Γ
.
\Gamma.
Γ.
Prove that the quadrilateral
A
P
N
Q
APNQ
A
PNQ
is a parallelogram.