MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea National Olympiad
2006 Korea National Olympiad
2006 Korea National Olympiad
Part of
Korea National Olympiad
Subcontests
(8)
8
1
Hide problems
27 students
27
27
27
students are given a number from
1
1
1
to
27.
27.
27.
How many ways are there to divide
27
27
27
students into
9
9
9
groups of
3
3
3
with the following condition?(i) The sum of students number in each group is
1
(
m
o
d
3
)
1\pmod{3}
1
(
mod
3
)
(ii) There are no such two students where their numbering differs by
3.
3.
3.
7
1
Hide problems
6 points on circle again
Points
A
,
B
,
C
,
D
,
E
,
F
A,B,C,D,E,F
A
,
B
,
C
,
D
,
E
,
F
is on the circle
O
.
O.
O
.
A line
ℓ
\ell
ℓ
is tangent to
O
O
O
at
E
E
E
is parallel to
A
C
AC
A
C
and
D
E
>
E
F
.
DE>EF.
D
E
>
EF
.
Let
P
,
Q
P,Q
P
,
Q
be the intersection of
ℓ
\ell
ℓ
and
B
C
,
C
D
BC,CD
BC
,
C
D
,respectively and let
R
,
S
R,S
R
,
S
be the intersection of
ℓ
\ell
ℓ
and
C
F
,
D
F
CF,DF
CF
,
D
F
,respectively. Show that
P
Q
=
R
S
PQ=RS
PQ
=
RS
if and only if
Q
E
=
E
R
.
QE=ER.
QE
=
ER
.
6
1
Hide problems
3-Vari Ineq
Prove that for any positive real numbers
x
,
y
x,y
x
,
y
and
z
,
z,
z
,
x
y
z
(
x
+
2
)
(
y
+
2
)
(
z
+
2
)
≤
(
1
+
2
(
x
y
+
y
z
+
z
x
)
3
)
3
xyz(x+2)(y+2)(z+2)\le(1+\frac{2(xy+yz+zx)}{3})^3
x
yz
(
x
+
2
)
(
y
+
2
)
(
z
+
2
)
≤
(
1
+
3
2
(
x
y
+
yz
+
z
x
)
)
3
5
1
Hide problems
Phi Function
Find all positive integers
n
n
n
such that
ϕ
(
n
)
\phi(n)
ϕ
(
n
)
is the fourth power of some prime.
4
1
Hide problems
Area is less than half
On the circle
O
,
O,
O
,
six points
A
,
B
,
C
,
D
,
E
,
F
A,B,C,D,E,F
A
,
B
,
C
,
D
,
E
,
F
are on the circle counterclockwise.
B
D
BD
B
D
is the diameter of the circle and it is perpendicular to
C
F
.
CF.
CF
.
Also, lines
C
F
,
B
E
,
A
D
CF, BE, AD
CF
,
BE
,
A
D
is concurrent. Let
M
M
M
be the foot of altitude from
B
B
B
to
A
C
AC
A
C
and let
N
N
N
be the foot of altitude from
D
D
D
to
C
E
.
CE.
CE
.
Prove that the area of
△
M
N
C
\triangle MNC
△
MNC
is less than half the area of
□
A
C
E
F
.
\square ACEF.
□
A
CEF
.
3
1
Hide problems
Classic NT
For three positive integers
a
,
b
a,b
a
,
b
and
c
,
c,
c
,
if
gcd
(
a
,
b
,
c
)
=
1
\text{gcd}(a,b,c)=1
gcd
(
a
,
b
,
c
)
=
1
and
a
2
+
b
2
+
c
2
=
2
(
a
b
+
b
c
+
c
a
)
,
a^2+b^2+c^2=2(ab+bc+ca),
a
2
+
b
2
+
c
2
=
2
(
ab
+
b
c
+
c
a
)
,
prove that all of
a
,
b
,
c
a,b,c
a
,
b
,
c
is perfect square.
2
1
Hide problems
Factoring Game
Alice and Bob are playing "factoring game." On the paper,
270000
(
=
2
4
3
3
5
4
)
270000(=2^43^35^4)
270000
(
=
2
4
3
3
5
4
)
is written and each person picks one number from the paper(call it
N
N
N
) and erase
N
N
N
and writes integer
X
,
Y
X,Y
X
,
Y
such that
N
=
X
Y
N=XY
N
=
X
Y
and
gcd
(
X
,
Y
)
≠
1.
\text{gcd}(X,Y)\ne1.
gcd
(
X
,
Y
)
=
1.
Alice goes first and the person who can no longer make this factoring loses. If two people use optimal strategy, prove that Alice always win.
1
1
Hide problems
Classic Algebra
Given that for reals
a
1
,
⋯
,
a
2004
,
a_1,\cdots, a_{2004},
a
1
,
⋯
,
a
2004
,
equation
x
2006
−
2006
x
2005
+
a
2004
x
2004
+
⋯
+
a
2
x
2
+
a
1
x
+
1
=
0
x^{2006}-2006x^{2005}+a_{2004}x^{2004}+\cdots +a_2x^2+a_1x+1=0
x
2006
−
2006
x
2005
+
a
2004
x
2004
+
⋯
+
a
2
x
2
+
a
1
x
+
1
=
0
has
2006
2006
2006
positive real solution, find the maximum possible value of
a
1
.
a_1.
a
1
.