MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2004 Korea Junior Math Olympiad
2004 Korea Junior Math Olympiad
Part of
Korea Junior Mathematics Olympiad
Subcontests
(5)
5
1
Hide problems
f:R->R s.t. f(f(x))-x^2+x+3=0
Show that there exists no function
f
:
R
→
R
f:\mathbb {R}\rightarrow \mathbb {R}
f
:
R
→
R
that satisfies
f
(
f
(
x
)
)
−
x
2
+
x
+
3
=
0
f(f(x))-x^2+x+3=0
f
(
f
(
x
))
−
x
2
+
x
+
3
=
0
for arbitrary real variable
x
x
x
.(Same as KMO 2004 P1)
4
1
Hide problems
A, B, and two excircles cyc
A
B
C
D
ABCD
A
BC
D
is a cyclic quadrilateral inscribed in circle
O
O
O
. Let
O
1
O_1
O
1
be the
A
A
A
-excenter of
A
B
C
ABC
A
BC
and
O
2
O_2
O
2
the
A
A
A
-excenter of
A
B
D
ABD
A
B
D
. Show that
A
,
B
,
O
1
,
O
2
A, B, O_1, O_2
A
,
B
,
O
1
,
O
2
is concyclic, and
O
O
O
passes through the center of
(
A
B
O
1
O
2
)
(ABO_1O_2)
(
A
B
O
1
O
2
)
. Recall that for concyclic
X
,
Y
,
Z
,
W
X, Y, Z, W
X
,
Y
,
Z
,
W
, the notation
(
X
Y
Z
W
)
(XYZW)
(
X
Y
Z
W
)
denotes the circumcircle of
X
Y
Z
W
XYZW
X
Y
Z
W
.
3
1
Hide problems
Multiples of p expressable as
For an arbitrary prime number
p
p
p
, show that there exists infinitely many multiples of
p
p
p
that can be expressed as the form
x
2
+
y
+
1
x
+
y
2
+
1
\frac{x^2+y+1}{x+y^2+1}
x
+
y
2
+
1
x
2
+
y
+
1
Where
x
,
y
x, y
x
,
y
are some positive integers.
2
1
Hide problems
Existence of a subset of S_n
For
n
≥
3
n\geq3
n
≥
3
define
S
n
=
{
1
,
2
,
.
.
.
,
n
}
S_n=\{1, 2, ..., n\}
S
n
=
{
1
,
2
,
...
,
n
}
.
A
1
,
A
2
,
.
.
.
,
A
n
A_1, A_{2}, ..., A_{n}
A
1
,
A
2
,
...
,
A
n
are given subsets of
S
n
S_n
S
n
, each having an even number of elements. Prove that there exists a set
{
i
1
,
i
2
,
.
.
.
,
i
t
}
\{i_1, i_2, ..., i_t\}
{
i
1
,
i
2
,
...
,
i
t
}
, a nonempty subset of
S
n
S_n
S
n
such that
A
i
1
Δ
A
i
2
Δ
…
Δ
A
i
t
=
∅
A_{i_1} \Delta A_{i_2} \Delta \ldots \Delta A_{i_t}=\emptyset
A
i
1
Δ
A
i
2
Δ
…
Δ
A
i
t
=
∅
(For two sets
A
,
B
A, B
A
,
B
, we define
Δ
\Delta
Δ
as
A
Δ
B
=
(
A
∪
B
)
−
(
A
∩
B
)
A \Delta B=(A\cup B)-(A\cap B)
A
Δ
B
=
(
A
∪
B
)
−
(
A
∩
B
)
)
1
1
Hide problems
Inscribing five squares
For positive reals
a
1
,
a
2
,
.
.
.
,
a
5
a_1, a_2, ..., a_5
a
1
,
a
2
,
...
,
a
5
such that
a
1
2
+
a
2
2
+
.
.
.
+
a
5
2
=
2
a^2_1+a^2_2+...+a^2_5=2
a
1
2
+
a
2
2
+
...
+
a
5
2
=
2
, consider five squares with sides
a
1
,
a
2
,
.
.
.
,
a
5
a_1, a_2, ..., a_5
a
1
,
a
2
,
...
,
a
5
respectively. Show that these squares can be placed inside (including boundaries) a square with side length of
2
2
2
so that the square themselves do not overlap each other.