MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea National Olympiad
2004 Korea National Olympiad
2004 Korea National Olympiad
Part of
Korea National Olympiad
Subcontests
(5)
4
1
Hide problems
KMO second round #4
Let
k
k
k
and
N
N
N
be positive real numbers which satisfy
k
≤
N
k\leq N
k
≤
N
. For
1
≤
i
≤
k
1\leq i \leq k
1
≤
i
≤
k
, there are subsets
A
i
A_i
A
i
of
{
1
,
2
,
3
,
…
,
N
}
\{1,2,3,\ldots,N\}
{
1
,
2
,
3
,
…
,
N
}
that satisfy the following property.For arbitrary subset of
{
i
1
,
i
2
,
…
,
i
s
}
⊂
{
1
,
2
,
3
,
…
,
k
}
\{ i_1, i_2, \ldots , i_s \} \subset \{ 1, 2, 3, \ldots, k \}
{
i
1
,
i
2
,
…
,
i
s
}
⊂
{
1
,
2
,
3
,
…
,
k
}
,
A
i
1
△
A
i
2
△
.
.
.
△
A
i
s
A_{i_1} \triangle A_{i_2} \triangle ... \triangle A_{i_s}
A
i
1
△
A
i
2
△...△
A
i
s
is not an empty set.Show that a subset
{
j
1
,
j
2
,
.
.
,
j
t
}
⊂
{
1
,
2
,
.
.
.
,
k
}
\{ j_1, j_2, .. ,j_t \} \subset \{ 1, 2, ... ,k \}
{
j
1
,
j
2
,
..
,
j
t
}
⊂
{
1
,
2
,
...
,
k
}
exist that satisfies
n
(
A
j
1
△
A
j
2
△
⋯
△
A
j
t
)
≥
k
n(A_{j_1} \triangle A_{j_2} \triangle \cdots \triangle A_{j_t}) \geq k
n
(
A
j
1
△
A
j
2
△
⋯
△
A
j
t
)
≥
k
. (
A
△
B
=
A
∪
B
−
A
∩
B
A \triangle B=A \cup B-A \cap B
A
△
B
=
A
∪
B
−
A
∩
B
)
5
1
Hide problems
KMO second round #5
A
,
B
,
C
A, B, C
A
,
B
,
C
, and
D
D
D
are the four different points on the circle
O
O
O
in the order. Let the centre of the scribed circle of triangle
A
B
C
ABC
A
BC
, which is tangent to
B
C
BC
BC
, be
O
1
O_1
O
1
. Let the centre of the scribed circle of triangle
A
C
D
ACD
A
C
D
, which is tangent to
C
D
CD
C
D
, be
O
2
O_2
O
2
.(1) Show that the circumcentre of triangle
A
B
O
1
ABO_1
A
B
O
1
is on the circle
O
O
O
.(2) Show that the circumcircle of triangle
C
O
1
O
2
CO_1O_2
C
O
1
O
2
always pass through a fixed point on the circle
O
O
O
, when
C
C
C
is moving along arc
B
D
BD
B
D
.
3
1
Hide problems
KMO second round #3
Positive real numbers,
a
1
,
.
.
,
a
6
a_1, .. ,a_6
a
1
,
..
,
a
6
satisfy
a
1
2
+
.
.
+
a
6
2
=
2
a_1^2+..+a_6^2=2
a
1
2
+
..
+
a
6
2
=
2
. Think six squares that has side length of
a
i
a_i
a
i
(
i
=
1
,
2
,
…
,
6
i=1,2,\ldots,6
i
=
1
,
2
,
…
,
6
). Show that the squares can be packed inside a square of length
2
2
2
, without overlapping.
2
1
Hide problems
KMO second round #2
x
x
x
and
y
y
y
are positive and relatively prime and
z
z
z
is an integer. They satisfy
(
5
z
−
4
x
)
(
5
z
−
4
y
)
=
25
x
y
(5z-4x)(5z-4y)=25xy
(
5
z
−
4
x
)
(
5
z
−
4
y
)
=
25
x
y
. Show that at least one of
10
z
+
x
+
y
10z+x+y
10
z
+
x
+
y
or quotient of this number divided by
3
3
3
is a square number (i.e. prove that
10
z
+
x
+
y
10z+x+y
10
z
+
x
+
y
or integer part of
10
z
+
x
+
y
3
\frac{10z+x+y}{3}
3
10
z
+
x
+
y
is a square number).
1
1
Hide problems
KMO second round #1
For arbitrary real number
x
x
x
, the function
f
:
R
→
R
f : \mathbb R \to \mathbb R
f
:
R
→
R
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
. Show that the function
f
f
f
does not exist.