MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
1998 Korea Junior Math Olympiad
1998 Korea Junior Math Olympiad
Part of
Korea Junior Mathematics Olympiad
Subcontests
(8)
8
1
Hide problems
1998 KJMO P8 Existence of 1998 elements in a set(T)
T
T
T
is a set of all the positive integers of the form
2
k
3
l
2^k 3^l
2
k
3
l
, where
k
,
l
k, l
k
,
l
are some non-negetive integers. Show that there exists
1998
1998
1998
different elements of
T
T
T
that satisfy the following condition.Condition The sum of the
1998
1998
1998
elements is again an element of
T
T
T
.
7
1
Hide problems
(1998 KJMO P7) ADE is an isosceles triangle
O
O
O
is a circumcircle of non-isosceles triangle
A
B
C
ABC
A
BC
and the angle bisector of
A
A
A
meets
B
C
BC
BC
at
D
D
D
. If the line perpendicular to
B
C
BC
BC
passing through
D
D
D
meets
A
O
AO
A
O
at
E
E
E
, show that
A
D
E
ADE
A
D
E
is an isosceles triangle.
6
1
Hide problems
(1998 KJMO P6) inequality on a/b+b/c+c/a
For positive reals
a
≥
b
≥
c
≥
0
a \geq b \geq c \geq 0
a
≥
b
≥
c
≥
0
prove the following inequality:
a
b
+
b
c
+
c
a
≥
a
+
b
a
+
c
+
b
+
c
b
+
a
+
c
+
a
c
+
b
\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \geq \frac{a+b}{a+c}+\frac{b+c}{b+a}+\frac{c+a}{c+b}
b
a
+
c
b
+
a
c
≥
a
+
c
a
+
b
+
b
+
a
b
+
c
+
c
+
b
c
+
a
5
1
Hide problems
complex geometry? 2n-gon (KJMO 1998 P5)
Regular
2
n
2n
2
n
-gon is inscribed in the unit circle. Find the sum of the squares of all sides and diagonal lengths in the
2
n
2n
2
n
-gon.
4
1
Hide problems
classic plane partition (KJMO 1998 P4)
n
n
n
lines are on the same plane, no two of them parallel and no three of them collinear(so the plane must be partitioned into some parts). How many parts is the plane partitioned into? Consider only the partitions with finitely large area.
3
1
Hide problems
KJMO 1998 P3 Homothety on orthocenter
O
O
O
is the circumcenter of
A
B
C
ABC
A
BC
, and
H
H
H
is the orthocenter of
A
B
C
ABC
A
BC
. If
D
D
D
is a midpoint of
A
C
AC
A
C
and
E
E
E
is the intersection of
B
O
BO
BO
and
A
B
C
ABC
A
BC
's circumcircle not
B
B
B
, show that three points
H
,
D
,
E
H, D, E
H
,
D
,
E
are collinear.
2
1
Hide problems
1998 KJMO P2 Graph theory Wire/Computer
There are
6
6
6
computers(power off) and
3
3
3
printers. Between a printer and a computer, they are connected with a wire or not. Printer can be only activated if and only if at least one of the connected computer's power is on. Your goal is to connect wires in such a way that, no matter how you choose three computers to turn on among the six, you can activate all
3
3
3
printers. What is the minimum number of wires required to make this possible?
1
1
Hide problems
1998 KJMO P1 Finding integer solutions(easy)
Show that there exist no integer solutions
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
to the equation
x
3
+
2
y
3
+
4
z
3
=
9
x^3+2y^3+4z^3=9
x
3
+
2
y
3
+
4
z
3
=
9