MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2006 Korea Junior Math Olympiad
2006 Korea Junior Math Olympiad
Part of
Korea Junior Mathematics Olympiad
Subcontests
(8)
8
1
Hide problems
Sigma_{i=1} ^{2006}a_ib_i = 0 where a_i, b_i in {-1,1}
De ne the set
F
F
F
as the following:
F
=
{
(
a
1
,
a
2
,
.
.
.
,
a
2006
)
:
∀
i
=
1
,
2
,
.
.
.
,
2006
,
a
i
∈
{
−
1
,
1
}
}
F = \{(a_1,a_2,... , a_{2006}) : \forall i = 1, 2,..., 2006, a_i \in \{-1,1\}\}
F
=
{(
a
1
,
a
2
,
...
,
a
2006
)
:
∀
i
=
1
,
2
,
...
,
2006
,
a
i
∈
{
−
1
,
1
}}
Prove that there exists a subset of
F
F
F
, called
S
S
S
which satis es the following:
∣
S
∣
=
2006
|S| = 2006
∣
S
∣
=
2006
and for all
(
a
1
,
a
2
,
.
.
.
,
a
2006
)
∈
F
(a_1,a_2,... , a_{2006})\in F
(
a
1
,
a
2
,
...
,
a
2006
)
∈
F
there exists
(
b
1
,
b
2
,
.
.
.
,
b
2006
)
∈
S
(b_1,b_2,... , b_{2006}) \in S
(
b
1
,
b
2
,
...
,
b
2006
)
∈
S
, such that
Σ
i
=
1
2006
a
i
b
i
=
0
\Sigma_{i=1} ^{2006}a_ib_i = 0
Σ
i
=
1
2006
a
i
b
i
=
0
.
4
1
Hide problems
transformation in coordinate plane, (a,b) ->(a + b, b), (a, b)-> (-b, a)
In the coordinate plane, de fine
M
=
{
(
a
,
b
)
,
a
,
b
∈
Z
}
M = \{(a, b),a,b \in Z\}
M
=
{(
a
,
b
)
,
a
,
b
∈
Z
}
. A transformation
S
S
S
, which is de fined on
M
M
M
, sends
(
a
,
b
)
(a,b)
(
a
,
b
)
to
(
a
+
b
,
b
)
(a + b, b)
(
a
+
b
,
b
)
. Transformation
T
T
T
, also de fined on
M
M
M
, sends
(
a
,
b
)
(a, b)
(
a
,
b
)
to
(
−
b
,
a
)
(-b, a)
(
−
b
,
a
)
. Prove that for all
(
a
,
b
)
∈
M
(a, b) \in M
(
a
,
b
)
∈
M
, we can use
S
,
T
S,T
S
,
T
denitely to map it to
(
g
,
0
)
(g,0)
(
g
,
0
)
.
1
1
Hide problems
\prod_{i = 1}^{2006} (a_{i}^2-i) is a mutliple of 3 when a_i is 1-2006
a
1
,
a
2
,
.
.
.
,
a
2006
a_1, a_2,...,a_{2006}
a
1
,
a
2
,
...
,
a
2006
is a permutation of
1
,
2
,
.
.
.
,
2006
1,2,...,2006
1
,
2
,
...
,
2006
. Prove that
∏
i
=
1
2006
(
a
i
2
−
i
)
\prod_{i = 1}^{2006} (a_{i}^2-i)
∏
i
=
1
2006
(
a
i
2
−
i
)
is a multiple of
3
3
3
. (
0
0
0
is counted as a multiple of
3
3
3
)
7
1
Hide problems
tangent line proof in KJMO 2006
A line through point
P
P
P
outside of circle
O
O
O
meets the said circle at
B
,
C
B,C
B
,
C
(
P
B
<
P
C
PB < PC
PB
<
PC
). Let
P
O
PO
PO
meet circle
O
O
O
at
Q
,
D
Q,D
Q
,
D
(with
P
Q
<
P
D
PQ < PD
PQ
<
P
D
). Let the line passing
Q
Q
Q
and perpendicular to
B
C
BC
BC
meet circle
O
O
O
at
A
A
A
. If
B
D
2
=
A
D
⋅
C
P
BD^2 = AD\cdot CP
B
D
2
=
A
D
⋅
CP
, prove that
P
A
PA
P
A
is a tangent to
O
O
O
.
6
1
Hide problems
\frac{a + b + c + d}{(1 + a^2)(1 + b^2)(1 + c^2)(1 + d^2)}< 1
For all reals
a
,
b
,
c
,
d
a, b, c,d
a
,
b
,
c
,
d
prove the following inequality:
a
+
b
+
c
+
d
(
1
+
a
2
)
(
1
+
b
2
)
(
1
+
c
2
)
(
1
+
d
2
)
<
1
\frac{a + b + c + d}{(1 + a^2)(1 + b^2)(1 + c^2)(1 + d^2)}< 1
(
1
+
a
2
)
(
1
+
b
2
)
(
1
+
c
2
)
(
1
+
d
2
)
a
+
b
+
c
+
d
<
1
5
1
Hide problems
(m^2 + 20mn + n^2(/(m^3 + n^3) pos. integer when m,n coprime pos. integers
Find all positive integers that can be written in the following way
m
2
+
20
m
n
+
n
2
m
3
+
n
3
\frac{m^2 + 20mn + n^2}{m^3 + n^3}
m
3
+
n
3
m
2
+
20
mn
+
n
2
Also,
m
,
n
m,n
m
,
n
are relatively prime positive integers.
2
1
Hide problems
b/a+c/a+c/b+a/b+a/c+b/c is an pos. integer, where a,b,c coprime pos. integers
Find all positive integers that can be written in the following way
b
a
+
c
a
+
c
b
+
a
b
+
a
c
+
b
c
\frac{b}{a}+\frac{c}{a}+\frac{c}{b}+\frac{a}{b}+\frac{a}{c}+\frac{b}{c}
a
b
+
a
c
+
b
c
+
b
a
+
c
a
+
c
b
. Also,
a
,
b
,
c
a,b, c
a
,
b
,
c
are positive integers that are pairwise relatively prime.
3
1
Hide problems
6 concyclic points given, concurrent, perpendicular, parallel wanted
In a circle
O
O
O
, there are six points,
A
,
B
,
C
,
D
,
E
,
F
A,B,C,D,E, F
A
,
B
,
C
,
D
,
E
,
F
in a counterclockwise order.
B
D
⊥
C
F
BD \perp CF
B
D
⊥
CF
, and
C
F
,
B
E
,
A
D
CF,BE,AD
CF
,
BE
,
A
D
are concurrent. Let the perpendicular from
B
B
B
to
A
C
AC
A
C
be
M
M
M
, and the perpendicular from
D
D
D
to
C
E
CE
CE
be
N
N
N
. Prove that
A
E
/
/
M
N
AE // MN
A
E
//
MN
.