MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea National Olympiad
2021 Korea National Olympiad
2021 Korea National Olympiad
Part of
Korea National Olympiad
Subcontests
(6)
P3
1
Hide problems
power of 2
Show that for any positive integers
k
k
k
and
1
≤
a
≤
9
1 \leq a \leq 9
1
≤
a
≤
9
, there exists
n
n
n
such that satisfies the below statement. When
2
n
=
a
0
+
10
a
1
+
1
0
2
a
2
+
⋯
+
1
0
i
a
i
+
⋯
2^n=a_0+10a_1+10^2a_2+ \cdots +10^ia_i+ \cdots
2
n
=
a
0
+
10
a
1
+
1
0
2
a
2
+
⋯
+
1
0
i
a
i
+
⋯
(
0
≤
a
i
≤
9
(0 \leq a_i \leq 9
(
0
≤
a
i
≤
9
and
a
i
a_i
a
i
is integer),
a
k
a_k
a
k
is equal to
a
a
a
.
P5
1
Hide problems
inequality& sequence
A real number sequence
a
1
,
⋯
,
a
2021
a_1, \cdots ,a_{2021}
a
1
,
⋯
,
a
2021
satisfies the below conditions.
a
1
=
1
,
a
2
=
2
,
a
n
+
2
=
2
a
n
+
1
2
a
n
+
a
n
+
1
(
1
≤
n
≤
2019
)
a_1=1, a_2=2, a_{n+2}=\frac{2a_{n+1}^2}{a_n+a_{n+1}} (1\leq n \leq 2019)
a
1
=
1
,
a
2
=
2
,
a
n
+
2
=
a
n
+
a
n
+
1
2
a
n
+
1
2
(
1
≤
n
≤
2019
)
Let the minimum of
a
1
,
⋯
,
a
2021
a_1, \cdots ,a_{2021}
a
1
,
⋯
,
a
2021
be
m
m
m
, and the maximum of
a
1
,
⋯
,
a
2021
a_1, \cdots ,a_{2021}
a
1
,
⋯
,
a
2021
be
M
M
M
. Let a 2021 degree polynomial
P
(
x
)
:
=
(
x
−
a
1
)
(
x
−
a
2
)
⋯
(
x
−
a
2021
)
P(x):=(x-a_1)(x-a_2) \cdots (x-a_{2021})
P
(
x
)
:=
(
x
−
a
1
)
(
x
−
a
2
)
⋯
(
x
−
a
2021
)
∣
P
(
x
)
∣
|P(x)|
∣
P
(
x
)
∣
is maximum in
[
m
,
M
]
[m, M]
[
m
,
M
]
when
x
=
α
x=\alpha
x
=
α
. Show that
1
<
α
<
2
1<\alpha <2
1
<
α
<
2
.
P4
1
Hide problems
Travel Route in a Bipartite Graph
For a positive integer
n
n
n
, there are two countries
A
A
A
and
B
B
B
with
n
n
n
airports each and
n
2
−
2
n
+
2
n^2-2n+ 2
n
2
−
2
n
+
2
airlines operating between the two countries. Each airline operates at least one flight. Exactly one flight by one of the airlines operates between each airport in
A
A
A
and each airport in
B
B
B
, and that flight operates in both directions. Also, there are no flights between two airports in the same country. For two different airports
P
P
P
and
Q
Q
Q
, denote by "
(
P
,
Q
)
(P, Q)
(
P
,
Q
)
-travel route" the list of airports
T
0
,
T
1
,
…
,
T
s
T_0, T_1, \ldots, T_s
T
0
,
T
1
,
…
,
T
s
satisfying the following conditions. [*]
T
0
=
P
,
T
s
=
Q
T_0=P,\ T_s=Q
T
0
=
P
,
T
s
=
Q
[*]
T
0
,
T
1
,
…
,
T
s
T_0, T_1, \ldots, T_s
T
0
,
T
1
,
…
,
T
s
are all distinct. [*] There exists an airline that operates between the airports
T
i
T_i
T
i
and
T
i
+
1
T_{i+1}
T
i
+
1
for all
i
=
0
,
1
,
…
,
s
−
1
i = 0, 1, \ldots, s-1
i
=
0
,
1
,
…
,
s
−
1
. Prove that there exist two airports
P
,
Q
P, Q
P
,
Q
such that there is no or exactly one
(
P
,
Q
)
(P, Q)
(
P
,
Q
)
-travel route. [hide=Graph Wording]Consider a complete bipartite graph
G
(
A
,
B
)
G(A, B)
G
(
A
,
B
)
with
∣
A
∣
=
∣
B
∣
=
n
\vert A \vert = \vert B \vert = n
∣
A
∣
=
∣
B
∣
=
n
. Suppose there are
n
2
−
2
n
+
2
n^2-2n+2
n
2
−
2
n
+
2
colors and each edge is colored by one of these colors. Define
(
P
,
Q
)
−
p
a
t
h
(P, Q)-path
(
P
,
Q
)
−
p
a
t
h
a path from
P
P
P
to
Q
Q
Q
such that all of the edges in the path are colored the same. Prove that there exist two vertices
P
P
P
and
Q
Q
Q
such that there is no or only one
(
P
,
Q
)
−
p
a
t
h
(P, Q)-path
(
P
,
Q
)
−
p
a
t
h
.
P2
1
Hide problems
Upgrade One-by-One Tuples Matching
For positive integers
n
,
k
,
r
n, k, r
n
,
k
,
r
, denote by
A
(
n
,
k
,
r
)
A(n, k, r)
A
(
n
,
k
,
r
)
the number of integer tuples
(
x
1
,
x
2
,
…
,
x
k
)
(x_1, x_2, \ldots, x_k)
(
x
1
,
x
2
,
…
,
x
k
)
satisfying the following conditions. [*]
x
1
≥
x
2
≥
⋯
≥
x
k
≥
0
x_1 \ge x_2 \ge \cdots \ge x_k \ge 0
x
1
≥
x
2
≥
⋯
≥
x
k
≥
0
[*]
x
1
+
x
2
+
⋯
+
x
k
=
n
x_1+x_2+ \cdots +x_k = n
x
1
+
x
2
+
⋯
+
x
k
=
n
[*]
x
1
−
x
k
≤
r
x_1-x_k \le r
x
1
−
x
k
≤
r
For all positive integers
m
,
s
,
t
m, s, t
m
,
s
,
t
, prove that
A
(
m
,
s
,
t
)
=
A
(
m
,
t
,
s
)
.
A(m, s, t)=A(m, t, s).
A
(
m
,
s
,
t
)
=
A
(
m
,
t
,
s
)
.
P6
1
Hide problems
Geometrical Geometry Problem
Let
A
B
C
ABC
A
BC
be an obtuse triangle with
∠
A
>
∠
B
>
∠
C
\angle A > \angle B > \angle C
∠
A
>
∠
B
>
∠
C
, and let
M
M
M
be a midpoint of the side
B
C
BC
BC
. Let
D
D
D
be a point on the arc
A
B
AB
A
B
of the circumcircle of triangle
A
B
C
ABC
A
BC
not containing
C
C
C
. Suppose that the circle tangent to
B
D
BD
B
D
at
D
D
D
and passing through
A
A
A
meets the circumcircle of triangle
A
B
M
ABM
A
BM
again at
E
E
E
and
B
D
‾
=
B
E
‾
\overline{BD}=\overline{BE}
B
D
=
BE
.
ω
\omega
ω
, the circumcircle of triangle
A
D
E
ADE
A
D
E
, meets
E
M
EM
EM
again at
F
F
F
. Prove that lines
B
D
BD
B
D
and
A
E
AE
A
E
meet on the line tangent to
ω
\omega
ω
at
F
F
F
.
P1
1
Hide problems
PQR Intersections
Let
A
B
C
ABC
A
BC
be an acute triangle and
D
D
D
be an intersection of the angle bisector of
A
A
A
and side
B
C
BC
BC
. Let
Ω
\Omega
Ω
be a circle tangent to the circumcircle of triangle
A
B
C
ABC
A
BC
and side
B
C
BC
BC
at
A
A
A
and
D
D
D
, respectively.
Ω
\Omega
Ω
meets the sides
A
B
,
A
C
AB, AC
A
B
,
A
C
again at
E
,
F
E, F
E
,
F
, respectively. The perpendicular line to
A
D
AD
A
D
, passing through
E
,
F
E, F
E
,
F
meets
Ω
\Omega
Ω
again at
G
,
H
G, H
G
,
H
, respectively. Suppose that
A
E
AE
A
E
and
G
D
GD
G
D
meet at
P
P
P
,
E
H
EH
E
H
and
G
F
GF
GF
meet at
Q
Q
Q
, and
H
D
HD
HD
and
A
F
AF
A
F
meet at
R
R
R
. Prove that
Q
F
‾
Q
G
‾
=
H
R
‾
P
G
‾
\dfrac{\overline{QF}}{\overline{QG}}=\dfrac{\overline{HR}}{\overline{PG}}
QG
QF
=
PG
H
R
.