MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea National Olympiad
2010 Korea National Olympiad
2010 Korea National Olympiad
Part of
Korea National Olympiad
Subcontests
(4)
4
2
Hide problems
Considering a simple graph with special conditions
There are
n
(
≥
4
)
n ( \ge 4 )
n
(
≥
4
)
people and some people shaked hands each other. Two people can shake hands at most 1 time. For arbitrary four people
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
such that
(
A
,
B
)
,
(
B
,
C
)
,
(
C
,
D
)
(A,B), (B,C), (C,D)
(
A
,
B
)
,
(
B
,
C
)
,
(
C
,
D
)
shaked hands, then one of
(
A
,
C
)
,
(
A
,
D
)
,
(
B
,
D
)
(A,C), (A,D), (B,D)
(
A
,
C
)
,
(
A
,
D
)
,
(
B
,
D
)
shaked hand each other. Prove the following statements. (a) Prove that
n
n
n
people can be divided into two groups,
X
,
Y
(
≠
∅
)
X, Y ( \ne \emptyset )
X
,
Y
(
=
∅
)
, such that for all
(
x
,
y
)
(x,y)
(
x
,
y
)
where
x
∈
X
x \in X
x
∈
X
and
y
∈
Y
y \in Y
y
∈
Y
,
x
x
x
and
y
y
y
shaked hands or
x
x
x
and
y
y
y
didn't shake hands. (b) There exist two people
A
,
B
A , B
A
,
B
such that the set of people who are not
A
A
A
and
B
B
B
that shaked hands with
A
A
A
is same wiith the set of people who are not
A
A
A
and
B
B
B
that shaked hands with
B
B
B
.
The remainder of 1+2+...+k
There are
2010
2010
2010
people sitting around a round table. First, we give one person
x
x
x
a candy. Next, we give candies to
1
1
1
st person,
1
+
2
1+2
1
+
2
th person,
1
+
2
+
3
1+2+3
1
+
2
+
3
th person,
⋯
\cdots
⋯
, and
1
+
2
+
⋯
+
2009
1+2+\cdots + 2009
1
+
2
+
⋯
+
2009
th person clockwise from
x
x
x
. Find the number of people who get at least one candy.
3
2
Hide problems
Three concurrent lines related to incircle
Let
I
I
I
be the incenter of triangle
A
B
C
ABC
A
BC
. The incircle touches
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
at points
P
,
Q
,
R
P, Q, R
P
,
Q
,
R
. A circle passing through
B
,
C
B , C
B
,
C
is tangent to the circle
I
I
I
at point
X
X
X
, a circle passing through
C
,
A
C , A
C
,
A
is tangent to the circle
I
I
I
at point
Y
Y
Y
, and a circle passing through
A
,
B
A , B
A
,
B
is tangent to the circle
I
I
I
at point
Z
Z
Z
, respectively. Prove that three lines
P
X
,
Q
Y
,
R
Z
PX, QY, RZ
PX
,
Q
Y
,
RZ
are concurrent.
A simple double counting on a simple graph
There are
2000
2000
2000
people, and some of them have called each other. Two people can call each other at most
1
1
1
time. For any two groups of three people
A
A
A
and
B
B
B
which
A
∩
B
=
∅
A \cap B = \emptyset
A
∩
B
=
∅
, there exist one person from each of
A
A
A
and
B
B
B
that haven't called each other. Prove that the number of two people called each other is less than
201000
201000
201000
.
2
2
Hide problems
3 variables with condition ab+bc+ca=1
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be positive real numbers such that
a
b
+
b
c
+
c
a
=
1
ab+bc+ca=1
ab
+
b
c
+
c
a
=
1
. Prove that
a
2
+
b
2
+
1
c
2
+
b
2
+
c
2
+
1
a
2
+
c
2
+
a
2
+
1
b
2
≥
33
\sqrt{ a^2 + b^2 + \frac{1}{c^2}} + \sqrt{ b^2 + c^2 + \frac{1}{a^2}} + \sqrt{ c^2 + a^2 + \frac{1}{b^2}} \ge \sqrt{33}
a
2
+
b
2
+
c
2
1
+
b
2
+
c
2
+
a
2
1
+
c
2
+
a
2
+
b
2
1
≥
33
Collinear points with orthocenters
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic convex quadrilateral. Let
E
E
E
be the intersection of lines
A
B
,
C
D
AB, CD
A
B
,
C
D
.
P
P
P
is the intersection of line passing
B
B
B
and perpendicular to
A
C
AC
A
C
, and line passing
C
C
C
and perpendicular to
B
D
BD
B
D
.
Q
Q
Q
is the intersection of line passing
D
D
D
and perpendicular to
A
C
AC
A
C
, and line passing
A
A
A
and perpendicular to
B
D
BD
B
D
. Prove that three points
E
,
P
,
Q
E, P, Q
E
,
P
,
Q
are collinear.
1
2
Hide problems
21 distinct prime divisors
Prove that
7
2
20
+
7
2
19
+
1
7^{2^{20}} + 7^{2^{19}} + 1
7
2
20
+
7
2
19
+
1
has at least
21
21
21
distinct prime divisors.
An inequality with x+y+z=1
x
,
y
,
z
x, y, z
x
,
y
,
z
are positive real numbers such that
x
+
y
+
z
=
1
x+y+z=1
x
+
y
+
z
=
1
. Prove that
x
1
−
x
+
y
1
−
y
+
z
1
−
z
>
2
\sqrt{ \frac{x}{1-x} } + \sqrt{ \frac{y}{1-y} } + \sqrt{ \frac{z}{1-z} } > 2
1
−
x
x
+
1
−
y
y
+
1
−
z
z
>
2