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Problems
Contests
National and Regional Contests
Korea Contests
Korea National Olympiad
2008 Korean National Olympiad
2008 Korean National Olympiad
Part of
Korea National Olympiad
Subcontests
(8)
8
1
Hide problems
2008 KMO P8
For fixed positive integers
s
,
t
s, t
s
,
t
, define
a
n
a_n
a
n
as the following.
a
1
=
s
,
a
2
=
t
a_1 = s, a_2 = t
a
1
=
s
,
a
2
=
t
, and
∀
n
≥
1
\forall n \ge 1
∀
n
≥
1
,
a
n
+
2
=
⌊
a
n
+
(
n
+
2
)
a
n
+
1
+
2008
⌋
a_{n+2} = \lfloor \sqrt{a_n+(n+2)a_{n+1}+2008} \rfloor
a
n
+
2
=
⌊
a
n
+
(
n
+
2
)
a
n
+
1
+
2008
⌋
. Prove that the solution set of
a
n
≠
n
a_n \not= n
a
n
=
n
,
n
∈
N
n \in \mathbb{N}
n
∈
N
is finite.
7
1
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2008 KMO P7
Prove that the only function
f
:
R
→
R
f: \mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
satisfying the following is
f
(
x
)
=
x
f(x)=x
f
(
x
)
=
x
. (i)
∀
x
≠
0
\forall x \not= 0
∀
x
=
0
,
f
(
x
)
=
x
2
f
(
1
x
)
f(x) = x^2f(\frac{1}{x})
f
(
x
)
=
x
2
f
(
x
1
)
. (ii)
∀
x
,
y
\forall x, y
∀
x
,
y
,
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
f(x+y) = f(x)+f(y)
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
. (iii)
f
(
1
)
=
1
f(1)=1
f
(
1
)
=
1
.
6
1
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2008 KMO P6
Let
A
B
C
D
ABCD
A
BC
D
be inscribed in a circle
ω
\omega
ω
. Let the line parallel to the tangent to
ω
\omega
ω
at
A
A
A
and passing
D
D
D
meet
ω
\omega
ω
at
E
E
E
.
F
F
F
is a point on
ω
\omega
ω
such that lies on the different side of
E
E
E
wrt
C
D
CD
C
D
. If
A
E
⋅
A
D
⋅
C
F
=
B
E
⋅
B
C
⋅
D
F
AE \cdot AD \cdot CF = BE \cdot BC \cdot DF
A
E
⋅
A
D
⋅
CF
=
BE
⋅
BC
⋅
D
F
and
∠
C
F
D
=
2
∠
A
F
B
\angle CFD = 2\angle AFB
∠
CF
D
=
2∠
A
FB
, Show that the tangent to
ω
\omega
ω
at
A
,
B
A, B
A
,
B
and line
E
F
EF
EF
concur at one point. (
A
A
A
and
E
E
E
lies on the same side of
C
D
CD
C
D
)
5
1
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2008 KMO P5
Let
p
p
p
be a prime where
p
≥
5
p \ge 5
p
≥
5
. Prove that
∃
n
\exists n
∃
n
such that
1
+
(
∑
i
=
2
n
1
i
2
)
(
∏
i
=
2
n
i
2
)
≡
0
(
m
o
d
p
)
1+ (\sum_{i=2}^n \frac{1}{i^2})(\prod_{i=2}^n i^2) \equiv 0 \pmod p
1
+
(
∑
i
=
2
n
i
2
1
)
(
∏
i
=
2
n
i
2
)
≡
0
(
mod
p
)
4
1
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2008 KMO P4
We define
A
,
B
,
C
A, B, C
A
,
B
,
C
as a partition of
N
\mathbb{N}
N
if
A
,
B
,
C
A,B,C
A
,
B
,
C
satisfies the following. (i)
A
,
B
,
C
≠
ϕ
A, B, C \not= \phi
A
,
B
,
C
=
ϕ
(ii)
A
∩
B
=
B
∩
C
=
C
∩
A
=
ϕ
A \cap B = B \cap C = C \cap A = \phi
A
∩
B
=
B
∩
C
=
C
∩
A
=
ϕ
(iii)
A
∪
B
∪
C
=
N
A \cup B \cup C = \mathbb{N}
A
∪
B
∪
C
=
N
.Prove that the partition of
N
\mathbb{N}
N
satisfying the following does not exist. (i)
∀
\forall
∀
a
∈
A
,
b
∈
B
a \in A, b \in B
a
∈
A
,
b
∈
B
, we have
a
+
b
+
2008
∈
C
a+b+2008 \in C
a
+
b
+
2008
∈
C
(ii)
∀
\forall
∀
b
∈
B
,
c
∈
C
b \in B, c \in C
b
∈
B
,
c
∈
C
, we have
b
+
c
+
2008
∈
A
b+c+2008 \in A
b
+
c
+
2008
∈
A
(iii)
∀
\forall
∀
c
∈
C
,
a
∈
A
c \in C, a \in A
c
∈
C
,
a
∈
A
, we have
c
+
a
+
2008
∈
B
c+a+2008 \in B
c
+
a
+
2008
∈
B
2
1
Hide problems
2008 KMO P2
We have
x
i
>
i
x_i >i
x
i
>
i
for all
1
≤
i
≤
n
1 \le i \le n
1
≤
i
≤
n
. Find the minimum value of
(
∑
i
=
1
n
x
i
)
2
∑
i
=
1
n
x
i
2
−
i
2
\frac{(\sum_{i=1}^n x_i)^2}{\sum_{i=1}^n \sqrt{x^2_i - i^2}}
∑
i
=
1
n
x
i
2
−
i
2
(
∑
i
=
1
n
x
i
)
2
1
1
Hide problems
2008 Korea P1
Let
V
=
[
(
x
,
y
,
z
)
∣
0
≤
x
,
y
,
z
≤
2008
]
V=[(x,y,z)|0\le x,y,z\le 2008]
V
=
[(
x
,
y
,
z
)
∣0
≤
x
,
y
,
z
≤
2008
]
be a set of points in a 3-D space. If the distance between two points is either
1
,
2
,
2
1, \sqrt{2}, 2
1
,
2
,
2
, we color the two points differently. How many colors are needed to color all points in
V
V
V
?
3
1
Hide problems
Tricky geo problem
Points
A
,
B
,
C
,
D
,
E
A,B,C,D,E
A
,
B
,
C
,
D
,
E
lie in a counterclockwise order on a circle
O
O
O
, and
A
C
=
C
E
AC = CE
A
C
=
CE
P
=
B
D
∩
A
C
P=BD \cap AC
P
=
B
D
∩
A
C
,
Q
=
B
D
∩
C
E
Q=BD \cap CE
Q
=
B
D
∩
CE
Let
O
1
O_1
O
1
be the circle which is tangent to
A
P
‾
,
B
P
‾
\overline {AP}, \overline {BP}
A
P
,
BP
and arc
A
B
AB
A
B
(which doesn't contain
C
C
C
) Let
O
2
O_2
O
2
be the circle which is tangent
D
Q
‾
,
E
Q
‾
\overline {DQ}, \overline {EQ}
D
Q
,
EQ
and arc
D
E
DE
D
E
(which doesn't contain
C
C
C
) Let
O
1
∩
O
=
R
,
O
2
∩
O
=
S
,
R
P
∩
Q
S
=
X
O_1 \cap O = R, O_2 \cap O = S, RP \cap QS = X
O
1
∩
O
=
R
,
O
2
∩
O
=
S
,
RP
∩
QS
=
X
Prove that
X
C
XC
XC
bisects
∠
A
C
E
\angle ACE
∠
A
CE