MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
1999 Korea Junior Math Olympiad
1999 Korea Junior Math Olympiad
Part of
Korea Junior Mathematics Olympiad
Subcontests
(8)
8
1
Hide problems
KJMO maximum value of m subsets
For
S
n
=
{
1
,
2
,
.
.
.
,
n
}
S_n=\{1, 2, ..., n\}
S
n
=
{
1
,
2
,
...
,
n
}
, find the maximum value of
m
m
m
that makes the following proposition true.Proposition There exists
m
m
m
different subsets of
S
S
S
, say
A
1
,
A
2
,
.
.
.
,
A
m
A_1, A_2, ..., A_m
A
1
,
A
2
,
...
,
A
m
, such that for every
i
,
j
=
1
,
2
,
.
.
.
,
m
i, j=1, 2, ..., m
i
,
j
=
1
,
2
,
...
,
m
, the set
A
i
∪
A
j
A_i \cup A_j
A
i
∪
A
j
is not
S
S
S
.
7
1
Hide problems
Maximum number of points place-able (KJMO 1999 P7)
A
0
B
,
A
0
C
A_0B, A_0C
A
0
B
,
A
0
C
rays that satisfy
∠
B
A
0
C
=
1
4
∘
\angle BA_0C=14^{\circ}
∠
B
A
0
C
=
1
4
∘
. You are to place points
A
1
,
A
2
,
.
.
.
A_1, A_2, ...
A
1
,
A
2
,
...
by the following rules. Rules (1) On the first move, place
A
1
A_1
A
1
on any point on
A
0
B
A_0B
A
0
B
(except
A
0
A_0
A
0
).(2) On the
n
>
1
n>1
n
>
1
th move, place
A
n
A_n
A
n
on
A
0
B
A_0B
A
0
B
iff
A
n
−
1
A_{n-1}
A
n
−
1
is on
A
0
C
A_0C
A
0
C
, and place
A
n
A_n
A
n
on
A
0
C
A_0C
A
0
C
iff
A
n
−
1
A_{n-1}
A
n
−
1
is one
A
0
B
A_0B
A
0
B
.
A
n
A_n
A
n
must be place on the point that satisfies
A
n
−
2
A
n
n
−
1
=
A
n
−
1
A
n
A_{n-2}A_n{n-1}=A_{n-1}A_n
A
n
−
2
A
n
n
−
1
=
A
n
−
1
A
n
.All the points must be placed in different locations. What is the maximum number of points that can be placed?
6
1
Hide problems
1999 KJMO P6 Smallest prime divisor of m
For a positive integer
n
n
n
, let
p
(
n
)
p(n)
p
(
n
)
denote the smallest prime divisor of
n
n
n
. Find the maximum number of divisors
m
m
m
can have if
p
(
m
)
4
>
m
p(m)^4>m
p
(
m
)
4
>
m
.
5
1
Hide problems
Easy circumcircle problem KJMO 1999 P5
O
O
O
is a circumcircle of
A
B
C
ABC
A
BC
and
C
O
CO
CO
meets
A
B
AB
A
B
at
P
P
P
, and
B
O
BO
BO
meets
A
C
AC
A
C
at
Q
Q
Q
. Show that
B
P
=
P
Q
=
Q
C
BP=PQ=QC
BP
=
PQ
=
QC
if and only if
∠
A
=
6
0
∘
\angle A=60^{\circ}
∠
A
=
6
0
∘
.
4
1
Hide problems
Trace of a square(Area)
C
C
C
is the unit circle in some plane.
R
R
R
is a square with side
a
a
a
.
C
C
C
is fixed and
R
R
R
moves(without rotation) on the plane, in such a way that its center stays inside
C
C
C
(including boundaries). Find the maximum value of the area drawn by the trace of
R
R
R
.
3
1
Hide problems
[a+\frac{b}{a}]=[b+\frac{a}{b}]
Recall that
[
x
]
[x]
[
x
]
denotes the largest integer not exceeding
x
x
x
for real
x
x
x
. For integers
a
,
b
a, b
a
,
b
in the interval
1
≤
a
<
b
≤
100
1 \leq a<b \leq 100
1
≤
a
<
b
≤
100
, find the number of ordered pairs
(
a
,
b
)
(a, b)
(
a
,
b
)
satisfying the following equation.
[
a
+
b
a
]
=
[
b
+
a
b
]
[a+\frac{b}{a}]=[b+\frac{a}{b}]
[
a
+
a
b
]
=
[
b
+
b
a
]
2
1
Hide problems
1999 KJMO sum, square sum, cubic sum
Three integers are given.
A
A
A
denotes the sum of the integers,
B
B
B
denotes the sum of the square of the integers and
C
C
C
denotes the sum of cubes of the integers(that is, if the three integers are
x
,
y
,
z
x, y, z
x
,
y
,
z
, then
A
=
x
+
y
+
z
A=x+y+z
A
=
x
+
y
+
z
,
B
=
x
2
+
y
2
+
z
2
B=x^2+y^2+z^2
B
=
x
2
+
y
2
+
z
2
,
C
=
x
3
+
y
3
+
z
3
C=x^3+y^3+z^3
C
=
x
3
+
y
3
+
z
3
). If
9
A
≥
B
+
60
9A \geq B+60
9
A
≥
B
+
60
and
C
≥
360
C \geq 360
C
≥
360
, find
A
,
B
,
C
A, B, C
A
,
B
,
C
.
1
1
Hide problems
Proving ABCD inscribed(only with strange conditions)
There exists point
O
O
O
inside a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
satisfying
O
A
=
O
B
OA=OB
O
A
=
OB
and
O
C
=
O
D
OC=OD
OC
=
O
D
, and
∠
A
O
B
=
∠
C
O
D
=
9
0
∘
\angle AOB = \angle COD=90^{\circ}
∠
A
OB
=
∠
CO
D
=
9
0
∘
. Consider two squares, (1)square having
A
C
AC
A
C
as one side and located in the opposite side of
B
B
B
and (2)square having
B
D
BD
B
D
as one side and located in the opposite side of
E
E
E
. If the common part of these two squares is also a square, prove that
A
B
C
D
ABCD
A
BC
D
is an inscribed quadrilateral.