MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea National Olympiad
2023 Korea National Olympiad
2023 Korea National Olympiad
Part of
Korea National Olympiad
Subcontests
(8)
6
1
Hide problems
Geometry, XYPQ concyclic
Let
Ω
\Omega
Ω
and
O
O
O
be the circumcircle and the circumcenter of an acute triangle
A
B
C
ABC
A
BC
(
A
B
‾
<
A
C
‾
)
(\overline{AB} < \overline{AC})
(
A
B
<
A
C
)
. Define
D
,
E
(
≠
A
)
D,E(\neq A)
D
,
E
(
=
A
)
be the points such that ray
A
O
AO
A
O
intersects
B
C
BC
BC
and
Ω
\Omega
Ω
. Let the line passing through
D
D
D
and perpendicular to
A
B
AB
A
B
intersects
A
C
AC
A
C
at
P
P
P
and define
Q
Q
Q
similarly. Tangents to
Ω
\Omega
Ω
on
A
,
E
A,E
A
,
E
intersects
B
C
BC
BC
at
X
,
Y
X,Y
X
,
Y
. Prove that
X
,
Y
,
P
,
Q
X,Y,P,Q
X
,
Y
,
P
,
Q
lie on a circle.
4
1
Hide problems
geometry in P4?
Pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
is inscribed in circle
Ω
\Omega
Ω
. Line
A
D
AD
A
D
meets
C
E
CE
CE
at
F
F
F
, and
P
(
≠
E
,
F
)
P (\neq E, F)
P
(
=
E
,
F
)
is a point on segment
E
F
EF
EF
. The circumcircle of triangle
A
F
P
AFP
A
FP
meets
Ω
\Omega
Ω
at
Q
(
≠
A
)
Q(\neq A)
Q
(
=
A
)
and
A
C
AC
A
C
at
R
(
≠
A
)
R(\neq A)
R
(
=
A
)
. Line
A
D
AD
A
D
meets
B
Q
BQ
BQ
at
S
S
S
, and the circumcircle of triangle
D
E
S
DES
D
ES
meets line
B
Q
,
B
D
BQ, BD
BQ
,
B
D
at
T
(
≠
S
)
,
U
(
≠
D
)
T(\neq S), U(\neq D)
T
(
=
S
)
,
U
(
=
D
)
, respectively. Prove that if
F
,
P
,
T
,
S
F, P, T, S
F
,
P
,
T
,
S
are concyclic, then
P
,
T
,
R
,
U
P, T, R, U
P
,
T
,
R
,
U
are concyclic.
3
1
Hide problems
Actually, estimating the size of A is harder than the problem
For a given positive integer
n
(
≥
2
)
n(\ge 2)
n
(
≥
2
)
, find maximum positive integer
A
A
A
such that there exists
P
∈
Z
[
x
]
P \in \mathbb{Z}[x]
P
∈
Z
[
x
]
with degree
n
n
n
that satisfies the following two conditions.[*] For any
1
≤
k
≤
A
1 \le k \le A
1
≤
k
≤
A
, it satisfies that
A
∣
P
(
k
)
A \mid P(k)
A
∣
P
(
k
)
, and [*]
P
(
0
)
=
0
P(0)= 0
P
(
0
)
=
0
and the coefficient of the first term of
P
P
P
is
1
1
1
, which means that
P
(
x
)
P(x)
P
(
x
)
is in the following form where
c
2
,
c
3
,
⋯
,
c
n
c_2, c_3, \cdots, c_n
c
2
,
c
3
,
⋯
,
c
n
are all integers and
c
n
≠
0
c_n \neq 0
c
n
=
0
.
P
(
x
)
=
c
n
x
n
+
c
n
−
1
x
n
−
1
+
⋯
+
c
2
x
2
+
x
P(x) = c_nx^n + c_{n-1}x^{n-1}+\dots+c_2x^2+x
P
(
x
)
=
c
n
x
n
+
c
n
−
1
x
n
−
1
+
⋯
+
c
2
x
2
+
x
7
1
Hide problems
a/b is unbounded
Positive real sequences
{
a
n
}
\{ a_n \}
{
a
n
}
and
{
b
n
}
\{ b_n \}
{
b
n
}
satisfy the following conditions for all positive integers
n
n
n
.[*]
a
n
+
1
b
n
+
1
=
a
n
2
+
b
n
2
a_{n+1}b_{n+1}= a_n^2 + b_n^2
a
n
+
1
b
n
+
1
=
a
n
2
+
b
n
2
[*]
a
n
+
1
+
b
n
+
1
=
a
n
b
n
a_{n+1}+b_{n+1}=a_nb_n
a
n
+
1
+
b
n
+
1
=
a
n
b
n
[*]
a
n
≥
b
n
a_n \geq b_n
a
n
≥
b
n
Prove that there exists positive integer
n
n
n
such that
a
n
b
n
>
202
3
2023
.
\frac{a_n}{b_n}>2023^{2023}.
b
n
a
n
>
202
3
2023
.
1
1
Hide problems
easy sequence
A sequence of positive reals
{
a
n
}
\{ a_n \}
{
a
n
}
is defined below.
a
0
=
1
,
a
1
=
3
,
a
n
+
2
=
a
n
+
1
2
+
2
a
n
a_0 = 1, a_1 = 3, a_{n+2} = \frac{a_{n+1}^2+2}{a_n}
a
0
=
1
,
a
1
=
3
,
a
n
+
2
=
a
n
a
n
+
1
2
+
2
Show that for all nonnegative integer
n
n
n
,
a
n
a_n
a
n
is a positive integer.
8
1
Hide problems
special number partition
For a positive integer
n
n
n
, if
n
n
n
is a product of two different primes and
n
≡
2
(
m
o
d
3
)
n \equiv 2 \pmod 3
n
≡
2
(
mod
3
)
, then
n
n
n
is called "special number." For example,
14
,
26
,
35
,
38
14, 26, 35, 38
14
,
26
,
35
,
38
is only special numbers among positive integers
1
1
1
to
50
50
50
. Prove that for any finite set
S
S
S
with special numbers, there exist two sets
A
,
B
A, B
A
,
B
such that[*]
A
∩
B
=
∅
,
A
∪
B
=
S
A \cap B = \emptyset, A \cup B = S
A
∩
B
=
∅
,
A
∪
B
=
S
[*]
∣
∣
A
∣
−
∣
B
∣
∣
≤
1
||A| - |B|| \leq 1
∣∣
A
∣
−
∣
B
∣∣
≤
1
[*] For all primes
p
p
p
, the difference between number of elements in
A
A
A
which is multiple of
p
p
p
and number of elements in
B
B
B
which is multiple of
p
p
p
is less than or equal to
1
1
1
.
5
1
Hide problems
phi sigma
Find all positive integers
n
n
n
such that
ϕ
(
n
)
+
σ
(
n
)
=
2
n
+
8.
\phi(n) + \sigma(n) = 2n + 8.
ϕ
(
n
)
+
σ
(
n
)
=
2
n
+
8.
2
1
Hide problems
x+2, x(x+1)/2
Sets
A
0
,
A
1
,
…
,
A
2023
A_0, A_1, \dots, A_{2023}
A
0
,
A
1
,
…
,
A
2023
satisfy the following conditions:[*]
A
0
=
{
3
}
A_0 = \{ 3 \}
A
0
=
{
3
}
[*]
A
n
=
{
x
+
2
∣
x
∈
A
n
−
1
}
∪
{
x
(
x
+
1
)
/
2
∣
x
∈
A
n
−
1
}
A_n = \{ x + 2 \mid x \in A_{n - 1} \} \ \cup \{x(x+1) / 2 \mid x \in A_{n - 1} \}
A
n
=
{
x
+
2
∣
x
∈
A
n
−
1
}
∪
{
x
(
x
+
1
)
/2
∣
x
∈
A
n
−
1
}
for each
n
=
1
,
2
,
…
,
2023
n = 1, 2, \dots, 2023
n
=
1
,
2
,
…
,
2023
.Find
∣
A
2023
∣
|A_{2023}|
∣
A
2023
∣
.