MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea National Olympiad
2009 Korea National Olympiad
2009 Korea National Olympiad
Part of
Korea National Olympiad
Subcontests
(4)
4
2
Hide problems
Double Counting on a simple Graph
There are
n
(
≥
3
)
n ( \ge 3)
n
(
≥
3
)
students in a class. Some students are friends each other, and friendship is always mutual. There are
s
(
≥
1
)
s ( \ge 1 )
s
(
≥
1
)
couples of two students who are friends, and
t
(
≥
1
)
t ( \ge 1 )
t
(
≥
1
)
triples of three students who are each friends. For two students
x
,
y
x, y
x
,
y
define
d
(
x
,
y
)
d(x,y)
d
(
x
,
y
)
be the number of students who are both friends with
x
x
x
and
y
y
y
. Prove that there exist three students
u
,
v
,
w
u, v, w
u
,
v
,
w
who are each friends and satisfying
d
(
u
,
v
)
+
d
(
v
,
w
)
+
d
(
w
,
u
)
≥
9
t
s
.
d(u,v) + d(v,w) + d(w,u) \ge \frac{9t}{s} .
d
(
u
,
v
)
+
d
(
v
,
w
)
+
d
(
w
,
u
)
≥
s
9
t
.
Minimum value of a function at a interval [0,n+1]
For a positive integer
n
n
n
, define a function
f
n
(
x
)
f_n (x)
f
n
(
x
)
at an interval
[
0
,
n
+
1
]
[ 0, n+1 ]
[
0
,
n
+
1
]
as
f
n
(
x
)
=
(
∑
i
=
1
n
∣
x
−
i
∣
)
2
−
∑
i
=
1
n
(
x
−
i
)
2
.
f_n (x) = ( \sum_{i=1} ^ {n} | x-i | )^2 - \sum_{i=1} ^{n} (x-i)^2 .
f
n
(
x
)
=
(
i
=
1
∑
n
∣
x
−
i
∣
)
2
−
i
=
1
∑
n
(
x
−
i
)
2
.
Let
a
n
a_n
a
n
be the minimum value of
f
n
(
x
)
f_n (x)
f
n
(
x
)
. Find the value of
∑
n
=
1
11
(
−
1
)
n
+
1
a
n
.
\sum_{n=1}^{11} (-1)^{n+1} a_n .
n
=
1
∑
11
(
−
1
)
n
+
1
a
n
.
3
2
Hide problems
Infinite descent
Let
n
n
n
be a positive integer. Suppose that the diophantine equation
z
n
=
8
x
2009
+
23
y
2009
z^n = 8 x^{2009} + 23 y^{2009}
z
n
=
8
x
2009
+
23
y
2009
uniquely has an integer solution
(
x
,
y
,
z
)
=
(
0
,
0
,
0
)
(x,y,z)=(0,0,0)
(
x
,
y
,
z
)
=
(
0
,
0
,
0
)
. Find the possible minimum value of
n
n
n
.
2^n-1 does not divide 3^n-1
For all positive integer
n
≥
2
n \ge 2
n
≥
2
, prove that
2
n
−
1
2^n -1
2
n
−
1
can't be a divisor of
3
n
−
1
3^n -1
3
n
−
1
.
2
2
Hide problems
Easy 3 variables inequality
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers. Prove that
a
3
c
(
a
2
+
b
c
)
+
b
3
a
(
b
2
+
c
a
)
+
c
3
b
(
c
2
+
a
b
)
≥
3
2
.
\frac{a^3}{c(a^2 + bc)} + \frac{b^3}{a(b^2 + ca)} + \frac{c^3}{b(c^2 +ab )} \ge \frac{3}{2} .
c
(
a
2
+
b
c
)
a
3
+
a
(
b
2
+
c
a
)
b
3
+
b
(
c
2
+
ab
)
c
3
≥
2
3
.
Convex quadrilateral
Let
A
B
C
ABC
A
BC
be a triangle and
P
,
Q
(
≠
A
,
B
,
C
)
P, Q ( \ne A, B, C )
P
,
Q
(
=
A
,
B
,
C
)
are the points lying on segments
B
C
,
C
A
BC , CA
BC
,
C
A
. Let
I
,
J
,
K
I, J, K
I
,
J
,
K
be the incenters of triangle
A
B
P
,
A
P
Q
,
C
P
Q
ABP, APQ, CPQ
A
BP
,
A
PQ
,
CPQ
. Prove that
P
I
J
K
PIJK
P
I
J
K
is a convex quadrilateral.
1
2
Hide problems
Homothety with incenter and circumcenters
Let
I
,
O
I, O
I
,
O
be the incenter and the circumcenter of triangle
A
B
C
ABC
A
BC
, and
D
,
E
,
F
D,E,F
D
,
E
,
F
be the circumcenters of triangle
B
I
C
,
C
I
A
,
A
I
B
BIC, CIA, AIB
B
I
C
,
C
I
A
,
A
I
B
. Let
P
,
Q
,
R
P, Q, R
P
,
Q
,
R
be the midpoints of segments
D
I
,
E
I
,
F
I
DI, EI, FI
D
I
,
E
I
,
F
I
. Prove that the circumcenter of triangle
P
Q
R
PQR
PQR
,
M
M
M
, is the midpoint of segment
I
O
IO
I
O
.
Number of one-to-one function
Let
A
=
{
1
,
2
,
3
,
⋯
,
12
}
A = \{ 1, 2, 3, \cdots , 12 \}
A
=
{
1
,
2
,
3
,
⋯
,
12
}
. Find the number of one-to-one function
f
:
A
→
A
f :A \to A
f
:
A
→
A
satisfying following condition: for all
i
∈
A
i \in A
i
∈
A
,
f
(
i
)
−
i
f(i)-i
f
(
i
)
−
i
is not a multiple of
3
3
3
.