MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2023 Korea Junior Math Olympiad
2023 Korea Junior Math Olympiad
Part of
Korea Junior Mathematics Olympiad
Subcontests
(8)
8
1
Hide problems
Covering equilateral triangle
A red equilateral triangle
T
T
T
with side length
1
1
1
is drawn on a plane. For a positive real
c
c
c
, we place three blue equilateral triangle shaped paper with side length
c
c
c
on a plane to cover
T
T
T
completely. Find the minimum value of
c
c
c
. As shown in the picture, it doesn't matter if the blue papers overlap each other or stick out from
T
T
T
. Folding or tearing the paper is not allowed.
5
1
Hide problems
Stone swap
For a positive integer
n
(
≥
5
)
n(\geq 5)
n
(
≥
5
)
, there are
n
n
n
white stones and
n
n
n
black stones (total
2
n
2n
2
n
stones) lined up in a row. The first
n
n
n
stones from the left are white, and the next
n
n
n
stones are black. \underbrace{\Circle \Circle \cdots \Circle}_n \underbrace{\CIRCLE \CIRCLE \cdots \CIRCLE}_n You can swap the stones by repeating the following operation.(Operation) Choose a positive integer
k
(
≤
2
n
−
5
)
k (\leq 2n - 5)
k
(
≤
2
n
−
5
)
, and swap
k
k
k
-th stone and
(
k
+
5
)
(k+5)
(
k
+
5
)
-th stone from the left.Find all positive integers
n
n
n
such that we can make first
n
n
n
stones to be black and the next
n
n
n
stones to be white in finite number of operations.
7
1
Hide problems
same size same sum
Find the smallest positive integer
N
N
N
such that there are no different sets
A
,
B
A, B
A
,
B
that satisfy the following conditions. (Here,
N
N
N
is not a power of
2
2
2
. That is,
N
≠
1
,
2
1
,
2
2
,
…
N \neq 1, 2^1, 2^2, \dots
N
=
1
,
2
1
,
2
2
,
…
.) [*]
A
,
B
⊆
{
1
,
2
1
,
2
2
,
2
3
,
…
,
2
2023
}
∪
{
N
}
A, B \subseteq \{1, 2^1, 2^2, 2^3, \dots, 2^{2023}\} \cup \{ N \}
A
,
B
⊆
{
1
,
2
1
,
2
2
,
2
3
,
…
,
2
2023
}
∪
{
N
}
[*]
∣
A
∣
=
∣
B
∣
≥
1
|A| = |B| \geq 1
∣
A
∣
=
∣
B
∣
≥
1
[*] Sum of elements in
A
A
A
and sum of elements in
B
B
B
are equal.
4
1
Hide problems
Tennis tournament
2023
2023
2023
players participated in a tennis tournament, and any two players played exactly one match. There was no draw in any match, and no player won all the other players. If a player
A
A
A
satisfies the following condition, let
A
A
A
be "skilled player".(Condition) For each player
B
B
B
who won
A
A
A
, there is a player
C
C
C
who won
B
B
B
and lost to
A
A
A
. It turned out there are exactly
N
(
≥
0
)
N(\geq 0)
N
(
≥
0
)
skilled player. Find the minimum value of
N
N
N
.
2
1
Hide problems
DFG = FCG
Quadrilateral
A
B
C
D
(
A
D
‾
<
B
C
‾
)
ABCD (\overline{AD} < \overline{BC})
A
BC
D
(
A
D
<
BC
)
is inscribed in a circle, and
H
(
≠
A
,
B
)
H(\neq A, B)
H
(
=
A
,
B
)
is a point on segment
A
B
.
AB.
A
B
.
The circumcircle of triangle
B
C
H
BCH
BC
H
meets
B
D
BD
B
D
at
E
(
≠
B
)
E(\neq B)
E
(
=
B
)
and line
H
E
HE
H
E
meets
A
D
AD
A
D
at
F
F
F
. The circle passes through
C
C
C
and tangent to line
B
D
BD
B
D
at
E
E
E
meets
E
F
EF
EF
at
G
(
≠
E
)
.
G(\neq E).
G
(
=
E
)
.
Prove that
∠
D
F
G
=
∠
F
C
G
.
\angle DFG = \angle FCG.
∠
D
FG
=
∠
FCG
.
6
1
Hide problems
Integer inequality
Find the maximum value of real number
A
A
A
such that
3
x
2
+
y
2
+
1
≥
A
(
x
2
+
x
y
+
x
)
3x^2 + y^2 + 1 \geq A(x^2 + xy + x)
3
x
2
+
y
2
+
1
≥
A
(
x
2
+
x
y
+
x
)
for all positive integers
x
,
y
.
x, y.
x
,
y
.
3
1
Hide problems
Sum of squares
Positive integers
a
1
,
a
2
,
…
,
a
2023
a_1, a_2, \dots, a_{2023}
a
1
,
a
2
,
…
,
a
2023
satisfy the following conditions.[*]
a
1
=
5
,
a
2
=
25
a_1 = 5, a_2 = 25
a
1
=
5
,
a
2
=
25
[*]
a
n
+
2
=
7
a
n
+
1
−
a
n
−
6
a_{n + 2} = 7a_{n+1}-a_n-6
a
n
+
2
=
7
a
n
+
1
−
a
n
−
6
for each
n
=
1
,
2
,
…
,
2021
n = 1, 2, \dots, 2021
n
=
1
,
2
,
…
,
2021
Prove that there exist integers
x
,
y
x, y
x
,
y
such that
a
2023
=
x
2
+
y
2
.
a_{2023} = x^2 + y^2.
a
2023
=
x
2
+
y
2
.
1
1
Hide problems
y^2 = x^3 + 2x^2 + 2x + 1
Find all integer pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
such that
y
2
=
x
3
+
2
x
2
+
2
x
+
1.
y^2 = x^3 + 2x^2 + 2x + 1.
y
2
=
x
3
+
2
x
2
+
2
x
+
1.