MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea National Olympiad
2014 Korea National Olympiad
2014 Korea National Olympiad
Part of
Korea National Olympiad
Subcontests
(4)
2
2
Hide problems
easy functional equation
Determine all the functions
f
:
R
→
R
f : \mathbb{R}\rightarrow\mathbb{R}
f
:
R
→
R
that satisfies the following.
f
(
x
f
(
x
)
+
f
(
x
)
f
(
y
)
+
y
−
1
)
=
f
(
x
f
(
x
)
+
x
y
)
+
y
−
1
f(xf(x)+f(x)f(y)+y-1)=f(xf(x)+xy)+y-1
f
(
x
f
(
x
)
+
f
(
x
)
f
(
y
)
+
y
−
1
)
=
f
(
x
f
(
x
)
+
x
y
)
+
y
−
1
Find the # of functions
How many one-to-one functions
f
:
{
1
,
2
,
⋯
,
9
}
→
{
1
,
2
,
⋯
,
9
}
f : \{1, 2, \cdots, 9\} \rightarrow \{1, 2, \cdots, 9\}
f
:
{
1
,
2
,
⋯
,
9
}
→
{
1
,
2
,
⋯
,
9
}
satisfy (i) and (ii)? (i)
f
(
1
)
>
f
(
2
)
f(1)>f(2)
f
(
1
)
>
f
(
2
)
,
f
(
9
)
<
9
f(9)<9
f
(
9
)
<
9
. (ii) For each
i
=
3
,
4
,
⋯
,
8
i=3, 4, \cdots, 8
i
=
3
,
4
,
⋯
,
8
, if
f
(
1
)
,
⋯
,
f
(
i
−
1
)
f(1), \cdots, f(i-1)
f
(
1
)
,
⋯
,
f
(
i
−
1
)
are smaller than
f
(
i
)
f(i)
f
(
i
)
, then
f
(
i
+
1
)
f(i+1)
f
(
i
+
1
)
is also smaller than
f
(
i
)
f(i)
f
(
i
)
.
1
2
Hide problems
square number
For
x
,
y
x, y
x
,
y
positive integers,
x
2
−
4
y
+
1
x^2-4y+1
x
2
−
4
y
+
1
is a multiple of
(
x
−
2
y
)
(
1
−
2
y
)
(x-2y)(1-2y)
(
x
−
2
y
)
(
1
−
2
y
)
. Prove that
∣
x
−
2
y
∣
|x-2y|
∣
x
−
2
y
∣
is a square number.
Nice Geometry
There is a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
which satisfies
∠
A
=
∠
D
\angle A=\angle D
∠
A
=
∠
D
. Let the midpoints of
A
B
,
A
D
,
C
D
AB, AD, CD
A
B
,
A
D
,
C
D
be
L
,
M
,
N
L,M,N
L
,
M
,
N
. Let's say the intersection point of
A
C
,
B
D
AC, BD
A
C
,
B
D
be
E
E
E
. Let's say point
F
F
F
which lies on
M
E
→
\overrightarrow{ME}
ME
satisfies
M
E
‾
×
M
F
‾
=
M
A
‾
2
\overline{ME}\times \overline{MF}=\overline{MA}^{2}
ME
×
MF
=
M
A
2
.Prove that
∠
L
F
M
=
∠
M
F
N
\angle LFM=\angle MFN
∠
L
FM
=
∠
MFN
. :)
4
2
Hide problems
Path through n stations
There is a city with
n
n
n
metro stations, each located at a vertex of a regular n-polygon. Metro Line 1 is a line which only connects two non-neighboring stations
A
A
A
and
B
B
B
. Metro Line 2 is a cyclic line which passes through all the stations in a shape of regular n-polygon. For each line metro can run in any direction, and
A
A
A
and
B
B
B
are the stations which one can transfer into other line. The line between two neighboring stations is called 'metro interval'. For each station there is one stationmaster, and there are at least one female stationmaster and one male stationmaster. If
n
n
n
is odd, prove that for any integer
k
k
k
(
0
<
k
<
n
)
(0<k<n)
(
0
<
k
<
n
)
there is a path that starts from a station with a male stationmaster and ends at a station with a female stationmaster, passing through
k
k
k
metro intervals.
function
Prove that there exists a function
f
:
N
→
N
f : \mathbb{N} \rightarrow \mathbb{N}
f
:
N
→
N
that satisfies the following (1)
{
f
(
n
)
:
n
∈
N
}
\{f(n) : n\in\mathbb{N}\}
{
f
(
n
)
:
n
∈
N
}
is a finite set; and (2) For nonzero integers
x
1
,
x
2
,
…
,
x
1000
x_1, x_2, \ldots, x_{1000}
x
1
,
x
2
,
…
,
x
1000
that satisfy
f
(
∣
x
1
∣
)
=
f
(
∣
x
2
∣
)
=
⋯
=
f
(
∣
x
1000
∣
)
f(\left|x_1\right|)=f(\left|x_2\right|)=\cdots=f(\left|x_{1000}\right|)
f
(
∣
x
1
∣
)
=
f
(
∣
x
2
∣
)
=
⋯
=
f
(
∣
x
1000
∣
)
, then
x
1
+
2
x
2
+
2
2
x
3
+
2
3
x
4
+
2
4
x
5
+
⋯
+
2
999
x
1000
≠
0
x_1+2x_2+2^2x_3+2^3x_4+2^4x_5+\cdots+2^{999}x_{1000}\ne 0
x
1
+
2
x
2
+
2
2
x
3
+
2
3
x
4
+
2
4
x
5
+
⋯
+
2
999
x
1000
=
0
.
3
2
Hide problems
Tangent lines and a tangent circle
A
B
AB
A
B
is a chord of
O
O
O
and
A
B
AB
A
B
is not a diameter of
O
O
O
. The tangent lines to
O
O
O
at
A
A
A
and
B
B
B
meet at
C
C
C
. Let
M
M
M
and
N
N
N
be the midpoint of the segments
A
C
AC
A
C
and
B
C
BC
BC
, respectively. A circle passing through
C
C
C
and tangent to
O
O
O
meets line
M
N
MN
MN
at
P
P
P
and
Q
Q
Q
. Prove that
∠
P
C
Q
=
∠
C
A
B
\angle PCQ = \angle CAB
∠
PCQ
=
∠
C
A
B
.
3 variable inequality, hard
Let
x
,
y
,
z
x, y, z
x
,
y
,
z
be the real numbers that satisfies the following.
(
x
−
y
)
2
+
(
y
−
z
)
2
+
(
z
−
x
)
2
=
8
,
x
3
+
y
3
+
z
3
=
1
(x-y)^2+(y-z)^2+(z-x)^2=8, x^3+y^3+z^3=1
(
x
−
y
)
2
+
(
y
−
z
)
2
+
(
z
−
x
)
2
=
8
,
x
3
+
y
3
+
z
3
=
1
Find the minimum value of
x
4
+
y
4
+
z
4
x^4+y^4+z^4
x
4
+
y
4
+
z
4
.