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Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2002 Korea Junior Math Olympiad
2002 Korea Junior Math Olympiad
Part of
Korea Junior Mathematics Olympiad
Subcontests
(8)
8
1
Hide problems
2003 KJMO P8
On a long metal stick,
1000
1000
1000
red marbles are embedded in the stick so the stick is equally partitioned into
1001
1001
1001
parts by them.
1001
1001
1001
blue marbles are embedded in the stick too, so the stick is equally partitioned into
1002
1002
1002
parts by them. If you cut the metal stick equally into
2003
2003
2003
smaller parts, how many of the smaller parts would contain at least one embedded marble?
7
1
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2002 KJMO P7 incenter one
I
I
I
is the incenter of
A
B
C
ABC
A
BC
.
D
D
D
is the intersection of
A
I
AI
A
I
and the circumcircle of
A
B
C
ABC
A
BC
, not
A
A
A
. And
P
P
P
is a midpoint of
B
I
BI
B
I
. If
C
I
=
2
A
I
CI=2AI
C
I
=
2
A
I
, show that
A
B
=
P
D
AB=PD
A
B
=
P
D
.
6
1
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2002 KJMO P6
For given positive integer
a
a
a
, find every
(
x
1
,
x
2
,
…
,
x
2002
)
(x_1, x_2, …, x_{2002})
(
x
1
,
x
2
,
…
,
x
2002
)
that satisfies the following:(1)
x
1
≥
x
2
≥
…
≥
x
2002
≥
0
x_1 \geq x_2 \geq … \geq x_{2002} \geq 0
x
1
≥
x
2
≥
…
≥
x
2002
≥
0
(2)
0
<
x
1
+
x
2
+
…
+
x
2003
<
a
+
1
0< x_1+x_2+…+x_{2003}<a+1
0
<
x
1
+
x
2
+
…
+
x
2003
<
a
+
1
(3)
x
1
2
+
x
2
2
+
…
+
x
2003
2
+
9
=
a
2
x^2_1+x^2_2+…+x^2_{2003}+9=a^2
x
1
2
+
x
2
2
+
…
+
x
2003
2
+
9
=
a
2
5
1
Hide problems
2002 KJMO integer solution
Find all integer solutions to the equation
x
3
+
2
y
3
+
4
z
3
+
8
x
y
z
=
0
x^3+2y^3+4z^3+8xyz=0
x
3
+
2
y
3
+
4
z
3
+
8
x
yz
=
0
4
1
Hide problems
2002 KJMO P4
For two non-negative integers
i
,
j
i, j
i
,
j
, create a new integer
i
#
j
i \# j
i
#
j
defined as the following: Express the two numbers in base
2
2
2
, and compare each digit. If their
k
k
k
th digit is the same, then the
k
k
k
th digit of
i
#
j
i \# j
i
#
j
is
0
0
0
. If their
k
k
k
th digit is different, then the
k
k
k
th digit of
i
#
j
i \# j
i
#
j
is
1
1
1
(of course we are talking in base
2
2
2
). For instance,
3
#
5
=
6
3 \# 5=6
3#5
=
6
. Show that for arbitrary positive integer
n
n
n
, the number can be expressed with finite operations of
#
\#
#
s and integers of the form
2
k
−
1
2^k-1
2
k
−
1
.
3
1
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2002 KJMO P3
For square
A
B
C
D
ABCD
A
BC
D
,
M
M
M
is a midpoint of segment
C
D
CD
C
D
and
E
E
E
is a point on
A
D
AD
A
D
satisfying
∠
B
E
M
=
∠
M
E
D
\angle BEM = \angle MED
∠
BEM
=
∠
ME
D
.
P
P
P
is an intersection of
A
M
AM
A
M
,
B
E
BE
BE
. Find the value of
P
E
B
P
\frac{PE}{BP}
BP
PE
2
1
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2002 KJMO P2
Find all prime number
p
p
p
such that
p
2002
+
200
3
p
−
1
−
1
p^{2002}+2003^{p-1}-1
p
2002
+
200
3
p
−
1
−
1
is a multiple of
2003
p
2003p
2003
p
.
1
1
Hide problems
2002 KJMO P1
Find the value of
x
2
+
y
2
+
z
2
x^2+y^2+z^2
x
2
+
y
2
+
z
2
where
x
,
y
,
z
x, y, z
x
,
y
,
z
are non-zero and satisfy the following:(1)
x
+
y
+
z
=
3
x+y+z=3
x
+
y
+
z
=
3
(2)
x
2
(
1
y
+
1
z
)
+
y
2
(
1
z
+
1
x
)
+
z
2
(
1
x
+
1
y
)
=
−
3
x^2(\frac{1}{y}+\frac{1}{z})+y^2(\frac{1}{z}+\frac{1}{x})+z^2(\frac{1}{x}+\frac{1}{y})=-3
x
2
(
y
1
+
z
1
)
+
y
2
(
z
1
+
x
1
)
+
z
2
(
x
1
+
y
1
)
=
−
3