MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea National Olympiad
2022 Korea National Olympiad
2022 Korea National Olympiad
Part of
Korea National Olympiad
Subcontests
(8)
6
1
Hide problems
Islands and the Bridges
n
(
≥
4
)
n(\geq 4)
n
(
≥
4
)
islands are connected by bridges to satisfy the following conditions: [*]Each bridge connects only two islands and does not go through other islands. [*]There is at most one bridge connecting any two different islands. [*]There does not exist a list
A
1
,
A
2
,
…
,
A
2
k
(
k
≥
2
)
A_1, A_2, \ldots, A_{2k}(k \geq 2)
A
1
,
A
2
,
…
,
A
2
k
(
k
≥
2
)
of distinct islands that satisfy the following: For every
i
=
1
,
2
,
…
,
2
k
i=1, 2, \ldots, 2k
i
=
1
,
2
,
…
,
2
k
, the two islands
A
i
A_i
A
i
and
A
i
+
1
A_{i+1}
A
i
+
1
are connected by a bridge. (Let
A
2
k
+
1
=
A
1
A_{2k+1}=A_1
A
2
k
+
1
=
A
1
)Prove that the number of the bridges is at most
3
(
n
−
1
)
2
\frac{3(n-1)}{2}
2
3
(
n
−
1
)
.
5
1
Hide problems
Tangent Length Inside the Incircle
For a scalene triangle
A
B
C
ABC
A
BC
with an incenter
I
I
I
, let its incircle meets the sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
at
D
,
E
,
F
D, E, F
D
,
E
,
F
, respectively. Denote by
P
P
P
the intersection of the lines
A
I
AI
A
I
and
D
F
DF
D
F
, and
Q
Q
Q
the intersection of the lines
B
I
BI
B
I
and
E
F
EF
EF
. Prove that
P
Q
‾
=
C
D
‾
\overline{PQ}=\overline{CD}
PQ
=
C
D
.
2
1
Hide problems
Parallel Lines of Circumcenters
In a scalene triangle
A
B
C
ABC
A
BC
, let the angle bisector of
A
A
A
meets side
B
C
BC
BC
at
D
D
D
. Let
E
,
F
E, F
E
,
F
be the circumcenter of the triangles
A
B
D
ABD
A
B
D
and
A
D
C
ADC
A
D
C
, respectively. Suppose that the circumcircles of the triangles
B
D
E
BDE
B
D
E
and
D
C
F
DCF
D
CF
intersect at
P
(
≠
D
)
P(\neq D)
P
(
=
D
)
, and denote by
O
,
X
,
Y
O, X, Y
O
,
X
,
Y
the circumcenters of the triangles
A
B
C
,
B
D
E
,
D
C
F
ABC, BDE, DCF
A
BC
,
B
D
E
,
D
CF
, respectively. Prove that
O
P
OP
OP
and
X
Y
XY
X
Y
are parallel.
7
1
Hide problems
Constant Sequence with Inequality
Suppose that the sequence
{
a
n
}
\{a_n\}
{
a
n
}
of positive reals satisfies the following conditions: [*]
a
i
≤
a
j
a_i \leq a_j
a
i
≤
a
j
for every positive integers
i
<
j
i <j
i
<
j
. [*]For any positive integer
k
≥
3
k \geq 3
k
≥
3
, the following inequality holds:
(
a
1
+
a
2
)
(
a
2
+
a
3
)
⋯
(
a
k
−
1
+
a
k
)
(
a
k
+
a
1
)
≤
(
2
k
+
2022
)
a
1
a
2
⋯
a
k
(a_1+a_2)(a_2+a_3)\cdots(a_{k-1}+a_k)(a_k+a_1)\leq (2^k+2022)a_1a_2\cdots a_k
(
a
1
+
a
2
)
(
a
2
+
a
3
)
⋯
(
a
k
−
1
+
a
k
)
(
a
k
+
a
1
)
≤
(
2
k
+
2022
)
a
1
a
2
⋯
a
k
Prove that
{
a
n
}
\{a_n\}
{
a
n
}
is constant.
3
1
Hide problems
Periodic Sequence with Perfect Square Condition
Suppose that the sequence
{
a
n
}
\{a_n\}
{
a
n
}
of positive integers satisfies the following conditions: [*]For an integer
i
≥
2022
i \geq 2022
i
≥
2022
, define
a
i
a_i
a
i
as the smallest positive integer
x
x
x
such that
x
+
∑
k
=
i
−
2021
i
−
1
a
k
x+\sum_{k=i-2021}^{i-1}a_k
x
+
∑
k
=
i
−
2021
i
−
1
a
k
is a perfect square. [*]There exists infinitely many positive integers
n
n
n
such that
a
n
=
4
×
2022
−
3
a_n=4\times 2022-3
a
n
=
4
×
2022
−
3
. Prove that there exists a positive integer
N
N
N
such that
∑
k
=
n
n
+
2021
a
k
\sum_{k=n}^{n+2021}a_k
∑
k
=
n
n
+
2021
a
k
is constant for every integer
n
≥
N
n \geq N
n
≥
N
. And determine the value of
∑
k
=
N
N
+
2021
a
k
\sum_{k=N}^{N+2021}a_k
∑
k
=
N
N
+
2021
a
k
.
1
1
Hide problems
Chained sequences
Three sequences
a
n
,
b
n
,
c
n
{a_n},{b_n},{c_n}
a
n
,
b
n
,
c
n
satisfy the following conditions.[*]
a
1
=
2
,
b
1
=
4
,
c
1
=
5
a_1=2,\,b_1=4,\,c_1=5
a
1
=
2
,
b
1
=
4
,
c
1
=
5
[*]
∀
n
,
a
n
+
1
=
b
n
+
1
c
n
,
b
n
+
1
=
c
n
+
1
a
n
,
c
n
+
1
=
a
n
+
1
b
n
\forall n,\; a_{n+1}=b_n+\frac{1}{c_n}, \, b_{n+1}=c_n+\frac{1}{a_n}, \, c_{n+1}=a_n+\frac{1}{b_n}
∀
n
,
a
n
+
1
=
b
n
+
c
n
1
,
b
n
+
1
=
c
n
+
a
n
1
,
c
n
+
1
=
a
n
+
b
n
1
Prove that for all positive integers
n
n
n
,
m
a
x
(
a
n
,
b
n
,
c
n
)
>
2
n
+
13
max(a_n,b_n,c_n)>\sqrt{2n+13}
ma
x
(
a
n
,
b
n
,
c
n
)
>
2
n
+
13
.
4
1
Hide problems
Cute counting
For positive integers
m
,
n
m, n
m
,
n
(
m
>
n
m>n
m
>
n
),
a
n
+
1
,
a
n
+
2
,
.
.
.
,
a
m
a_{n+1}, a_{n+2}, ..., a_m
a
n
+
1
,
a
n
+
2
,
...
,
a
m
are non-negative integers that satisfy the following inequality.
2
>
a
n
+
1
n
+
1
≥
a
n
+
2
n
+
2
≥
⋯
≥
a
m
m
2> \frac{a_{n+1}}{n+1} \ge \frac{a_{n+2}}{n+2} \ge \cdots \ge \frac{a_m}{m}
2
>
n
+
1
a
n
+
1
≥
n
+
2
a
n
+
2
≥
⋯
≥
m
a
m
Find the number of pair
(
a
n
+
1
,
a
n
+
2
,
⋯
,
a
m
)
(a_{n+1}, a_{n+2}, \cdots, a_m)
(
a
n
+
1
,
a
n
+
2
,
⋯
,
a
m
)
.
8
1
Hide problems
Equation on rational numbers
p
p
p
is a prime number such that its remainder divided by 8 is 3. Find all pairs of rational numbers
(
x
,
y
)
(x,y)
(
x
,
y
)
that satisfy the following equation.
p
2
x
4
−
6
p
x
2
+
1
=
y
2
p^2 x^4-6px^2+1=y^2
p
2
x
4
−
6
p
x
2
+
1
=
y
2