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Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2008 Korea Junior Math Olympiad
2008 Korea Junior Math Olympiad
Part of
Korea Junior Mathematics Olympiad
Subcontests
(8)
6
1
Hide problems
8 \(n^3f_1(n) - 2nf_9(n) + n^2f_3(n)) where, f_s(n) = d_1^s+d_2^s+..+d_k^s
If
d
1
,
d
2
,
.
.
.
,
d
k
d_1,d_2,...,d_k
d
1
,
d
2
,
...
,
d
k
are all distinct positive divisors of
n
n
n
, we defi ne
f
s
(
n
)
=
d
1
s
+
d
2
s
+
.
.
+
d
k
s
f_s(n) = d_1^s+d_2^s+..+d_k^s
f
s
(
n
)
=
d
1
s
+
d
2
s
+
..
+
d
k
s
. For example, we have
f
1
(
3
)
=
1
+
3
=
4
,
f
2
(
4
)
=
1
+
2
2
+
4
2
=
21
f_1(3) = 1 + 3 = 4, f_2(4) = 1 + 2^2 + 4^2 = 21
f
1
(
3
)
=
1
+
3
=
4
,
f
2
(
4
)
=
1
+
2
2
+
4
2
=
21
. Prove that for all positive integers
n
n
n
,
n
3
f
1
(
n
)
−
2
n
f
9
(
n
)
+
n
2
f
3
(
n
)
n^3f_1(n) - 2nf_9(n) + n^2f_3(n)
n
3
f
1
(
n
)
−
2
n
f
9
(
n
)
+
n
2
f
3
(
n
)
is divisible by
8
8
8
.
5
1
Hide problems
two tangents of the same circle and another line are concurrent, equal products
Let there be a pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
inscribed in a circle
O
O
O
. The tangent to
O
O
O
at
E
E
E
is parallel to
A
D
AD
A
D
. A point
F
F
F
lies on
O
O
O
and it is in the opposite side of
A
A
A
with respect to
C
D
CD
C
D
, and satisfi es
A
B
⋅
B
C
⋅
D
F
=
A
E
⋅
E
D
⋅
C
F
AB \cdot BC \cdot DF = AE \cdot ED \cdot CF
A
B
⋅
BC
⋅
D
F
=
A
E
⋅
E
D
⋅
CF
and
∠
C
F
D
=
2
∠
B
F
E
\angle CFD = 2\angle BFE
∠
CF
D
=
2∠
BFE
. Prove that the tangent to
O
O
O
at
B
,
E
B,E
B
,
E
and line
A
F
AF
A
F
concur at one point.
1
1
Hide problems
collinear points given product of 4 ratios =1
In a
△
X
Y
Z
\triangle XYZ
△
X
Y
Z
, points
A
,
B
A,B
A
,
B
lie on segment
Z
X
,
C
,
D
ZX, C,D
ZX
,
C
,
D
lie on segment
X
Y
,
E
,
F
XY , E, F
X
Y
,
E
,
F
lie on segment
Y
Z
YZ
Y
Z
.
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
lie on a circle, and
A
Z
⋅
E
Y
⋅
Z
B
⋅
Y
F
E
Z
⋅
C
Y
⋅
Z
F
⋅
Y
D
=
1
\frac{AZ \cdot EY \cdot ZB \cdot Y F}{EZ \cdot CY \cdot ZF \cdot Y D}= 1
EZ
⋅
C
Y
⋅
ZF
⋅
Y
D
A
Z
⋅
E
Y
⋅
ZB
⋅
Y
F
=
1
. Let
L
=
Z
X
∩
D
E
L = ZX \cap DE
L
=
ZX
∩
D
E
,
M
=
X
Y
∩
A
F
M = XY \cap AF
M
=
X
Y
∩
A
F
,
N
=
Y
Z
∩
B
C
N = Y Z \cap BC
N
=
Y
Z
∩
BC
. Prove that
L
,
M
,
N
L,M,N
L
,
M
,
N
are collinear.
7
1
Hide problems
f(x + y) = g (1/x+1/y) (xy)^{2008} , xy\ne 0 , find f,g:R -> R
Find all pairs of functions
f
;
g
:
R
→
R
f; g : R \to R
f
;
g
:
R
→
R
such that for all reals
x
.
y
≠
0
x.y \ne 0
x
.
y
=
0
:
f
(
x
+
y
)
=
g
(
1
x
+
1
y
)
⋅
(
x
y
)
2008
f(x + y) = g \left(\frac{1}{x}+\frac{1}{y}\right) \cdot (xy)^{2008}
f
(
x
+
y
)
=
g
(
x
1
+
y
1
)
⋅
(
x
y
)
2008
8
1
Hide problems
small groups created by 12 members in a club
There are
12
12
12
members in a club. The members created some small groups, which satisfy the following: - The small group consists of
3
3
3
or
4
4
4
people. - Also, for two arbitrary members, there exists exactly one small group that has both members. Prove that all members are in the same number of small groups.
4
1
Hide problems
partitions of N
Let
N
N
N
be the set of positive integers. If
A
,
B
,
C
≠
∅
A,B,C \ne \emptyset
A
,
B
,
C
=
∅
,
A
∩
B
=
B
∩
C
=
C
∩
A
=
∅
A \cap B = B \cap C = C \cap A = \emptyset
A
∩
B
=
B
∩
C
=
C
∩
A
=
∅
and
A
∪
B
∪
C
=
N
A \cup B \cup C = N
A
∪
B
∪
C
=
N
, we say that
A
,
B
,
C
A,B,C
A
,
B
,
C
are partitions of
N
N
N
. Prove that there are no partitions of
N
,
A
,
B
,
C
N, A,B,C
N
,
A
,
B
,
C
, that satis fy the following: (i)
∀
a
∈
A
,
b
∈
B
\forall a \in A, b \in B
∀
a
∈
A
,
b
∈
B
, we have
a
+
b
+
1
∈
C
a + b + 1 \in C
a
+
b
+
1
∈
C
(ii)
∀
b
∈
B
,
c
∈
C
\forall b \in B, c \in C
∀
b
∈
B
,
c
∈
C
, we have
b
+
c
+
1
∈
A
b + c + 1 \in A
b
+
c
+
1
∈
A
(iii)
∀
c
∈
C
,
a
∈
A
\forall c \in C, a \in A
∀
c
∈
C
,
a
∈
A
, we have
c
+
a
+
1
∈
B
c + a + 1 \in B
c
+
a
+
1
∈
B
3
1
Hide problems
x, y relatively prime to 5 such that x^2 + y^2 = 5^n for any n
For all positive integers
n
n
n
, prove that there are integers
x
,
y
x, y
x
,
y
relatively prime to
5
5
5
such that
x
2
+
y
2
=
5
n
x^2 + y^2 = 5^n
x
2
+
y
2
=
5
n
.
2
1
Hide problems
Find the mimimum value
Let
x
,
y
∈
R
x,y\in\mathbb{R}
x
,
y
∈
R
such that
x
>
2
,
y
>
3
x>2, y>3
x
>
2
,
y
>
3
. Find the minimum value of
(
x
+
y
)
2
x
2
−
4
+
y
2
−
9
\frac{(x+y)^2}{\sqrt{x^2-4}+\sqrt{y^2-9}}
x
2
−
4
+
y
2
−
9
(
x
+
y
)
2