MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea National Olympiad
2020 Korea National Olympiad
2020 Korea National Olympiad
Part of
Korea National Olympiad
Subcontests
(6)
6
1
Hide problems
Cute Geo
Let
A
B
C
D
E
ABCDE
A
BC
D
E
be a convex pentagon such that quadrilateral
A
B
D
E
ABDE
A
B
D
E
is a parallelogram and quadrilateral
B
C
D
E
BCDE
BC
D
E
is inscribed in a circle. The circle with center
C
C
C
and radius
C
D
CD
C
D
intersects the line
B
D
,
D
E
BD, DE
B
D
,
D
E
at points
F
,
G
(
≠
D
)
F, G(\neq D)
F
,
G
(
=
D
)
, and points
A
,
F
,
G
A, F, G
A
,
F
,
G
is on line l. Let
H
H
H
be the intersection point of line
l
l
l
and segment
B
C
BC
BC
. Consider the set of circle
Ω
\Omega
Ω
satisfying the following condition.Circle
Ω
\Omega
Ω
passes through
A
,
H
A, H
A
,
H
and intersects the sides
A
B
,
A
E
AB, AE
A
B
,
A
E
at point other than
A
A
A
.Let
P
,
Q
(
≠
A
)
P, Q(\neq A)
P
,
Q
(
=
A
)
be the intersection point of circle
Ω
\Omega
Ω
and sides
A
B
,
A
E
AB, AE
A
B
,
A
E
. Prove that
A
P
+
A
Q
AP+AQ
A
P
+
A
Q
is constant.
5
1
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Tricky Sequence
For some positive integer
n
n
n
, there exists
n
n
n
different positive integers
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2, ..., a_n
a
1
,
a
2
,
...
,
a
n
such that
(
1
)
(1)
(
1
)
a
1
=
1
,
a
n
=
2000
a_1=1, a_n=2000
a
1
=
1
,
a
n
=
2000
(
2
)
(2)
(
2
)
∀
i
∈
Z
\forall i\in \mathbb{Z}
∀
i
∈
Z
s
.
t
.
s.t.
s
.
t
.
2
≤
i
≤
n
,
a
i
−
a
i
−
1
∈
{
−
3
,
5
}
2\le i\le n, a_i -a_{i-1}\in \{-3,5\}
2
≤
i
≤
n
,
a
i
−
a
i
−
1
∈
{
−
3
,
5
}
Determine the maximum value of n.
3
1
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High school combinatorics
There are n boys and m girls at Daehan Mathematical High School. Let
d
(
B
)
d(B)
d
(
B
)
a number of girls who know Boy
B
B
B
each other, and let
d
(
G
)
d(G)
d
(
G
)
a number of boys who know Girl
G
G
G
each other. Each girl knows at least one boy each other. Prove that there exist Boy
B
B
B
and Girl
G
G
G
who knows each other in condition that
d
(
B
)
d
(
G
)
≥
m
n
\frac{d(B)}{d(G)}\ge\frac{m}{n}
d
(
G
)
d
(
B
)
≥
n
m
.
4
1
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m^2-5n^2 and m^2+5n^2 are both perfect squares
Find a pair of coprime positive integers
(
m
,
n
)
(m,n)
(
m
,
n
)
other than
(
41
,
12
)
(41,12)
(
41
,
12
)
such that
m
2
−
5
n
2
m^2-5n^2
m
2
−
5
n
2
and
m
2
+
5
n
2
m^2+5n^2
m
2
+
5
n
2
are both perfect squares.
2
1
Hide problems
P,Q,B are collinear
H
H
H
is the orthocenter of an acute triangle
A
B
C
ABC
A
BC
, and let
M
M
M
be the midpoint of
B
C
BC
BC
. Suppose
(
A
H
)
(AH)
(
A
H
)
meets
A
B
AB
A
B
and
A
C
AC
A
C
at
D
,
E
D,E
D
,
E
respectively.
A
H
AH
A
H
meets
D
E
DE
D
E
at
P
P
P
, and the line through
H
H
H
perpendicular to
A
H
AH
A
H
meets
D
M
DM
D
M
at
Q
Q
Q
. Prove that
P
,
Q
,
B
P,Q,B
P
,
Q
,
B
are collinear.
1
1
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Another FE
Determine all functions
f
:
R
→
R
f:\mathbb{R}\rightarrow\mathbb{R}
f
:
R
→
R
such that
x
2
f
(
x
)
+
y
f
(
y
2
)
=
f
(
x
+
y
)
f
(
x
2
−
x
y
+
y
2
)
x^2f(x)+yf(y^2)=f(x+y)f(x^2-xy+y^2)
x
2
f
(
x
)
+
y
f
(
y
2
)
=
f
(
x
+
y
)
f
(
x
2
−
x
y
+
y
2
)
for all
x
,
y
∈
R
x,y\in\mathbb{R}
x
,
y
∈
R
.