MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2022 Korea Junior Math Olympiad
2022 Korea Junior Math Olympiad
Part of
Korea Junior Mathematics Olympiad
Subcontests
(8)
2
1
Hide problems
normal counting problem in KJMO 2022
For positive integer
n
n
n
(≥
3
3
3
), find the number of ordered pairs
(
a
1
,
a
2
,
.
.
.
,
a
n
)
(a_1, a_2, ... , a_n)
(
a
1
,
a
2
,
...
,
a
n
)
of integers that satisfy the following two conditions1. For positive integer
i
i
i
such that
1
1
1
≤
i
i
i
≤
n
n
n
then
1
1
1
≤
a
i
a_i
a
i
≤
i
i
i
2. For positive integers
i
,
j
,
k
i,j,k
i
,
j
,
k
such that
1
1
1
≤
i
i
i
<
j
j
j
<
k
k
k
≤
n
n
n
, if
a
i
=
a
j
a_i = a_j
a
i
=
a
j
then
a
j
a_j
a
j
≥
a
k
a_k
a
k
6
1
Hide problems
Concyclic with a Tangent Circle
Let
A
B
C
ABC
A
BC
be a isosceles triangle with
A
B
‾
=
A
C
‾
\overline{AB}=\overline{AC}
A
B
=
A
C
. Let
D
(
≠
A
,
C
)
D(\neq A, C)
D
(
=
A
,
C
)
be a point on the side
A
C
AC
A
C
, and circle
Ω
\Omega
Ω
is tangent to
B
D
BD
B
D
at point
E
E
E
, and
A
C
AC
A
C
at point
C
C
C
. Denote by
F
(
≠
E
)
F(\neq E)
F
(
=
E
)
the intersection of the line
A
E
AE
A
E
and the circle
Ω
\Omega
Ω
, and
G
(
≠
a
)
G(\neq a)
G
(
=
a
)
the intersection of the line
A
C
AC
A
C
and the circumcircle of the triangle
A
B
F
ABF
A
BF
. Prove that points
D
,
E
,
F
,
D, E, F,
D
,
E
,
F
,
and
G
G
G
are concyclic.
3
1
Hide problems
Infinite in the Sequence of Remainder
For a given odd prime number
p
p
p
, define
f
(
n
)
f(n)
f
(
n
)
the remainder of
d
d
d
divided by
p
p
p
, where
d
d
d
is the biggest divisor of
n
n
n
which is not a multiple of
p
p
p
. For example when
p
=
5
p=5
p
=
5
,
f
(
6
)
=
1
,
f
(
35
)
=
2
,
f
(
75
)
=
3
f(6)=1, f(35)=2, f(75)=3
f
(
6
)
=
1
,
f
(
35
)
=
2
,
f
(
75
)
=
3
. Define the sequence
a
1
,
a
2
,
…
,
a
n
,
…
a_1, a_2, \ldots, a_n, \ldots
a
1
,
a
2
,
…
,
a
n
,
…
of integers as the followings:[*]
a
1
=
1
a_1=1
a
1
=
1
[*]
a
n
+
1
=
a
n
+
(
−
1
)
f
(
n
)
+
1
a_{n+1}=a_n+(-1)^{f(n)+1}
a
n
+
1
=
a
n
+
(
−
1
)
f
(
n
)
+
1
for all positive integers
n
n
n
. Determine all integers
m
m
m
, such that there exist infinitely many positive integers
k
k
k
such that
m
=
a
k
m=a_k
m
=
a
k
.
8
1
Hide problems
Rational Number Theory
Find all pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
of rational numbers such that
x
y
2
=
x
2
+
2
x
−
3
xy^2=x^2+2x-3
x
y
2
=
x
2
+
2
x
−
3
4
1
Hide problems
A strange distcrete functional equation seems analytic
Find all function
f
:
N
⟶
N
f:\mathbb{N} \longrightarrow \mathbb{N}
f
:
N
⟶
N
such thatforall positive integers
x
x
x
and
y
y
y
,
f
(
x
+
y
)
−
f
(
x
)
f
(
y
)
\frac{f(x+y)-f(x)}{f(y)}
f
(
y
)
f
(
x
+
y
)
−
f
(
x
)
is again a positive integer not exceeding
202
2
2022
2022^{2022}
202
2
2022
.
5
1
Hide problems
A strange recurrence relation
A sequence of real numbers
a
1
,
a
2
,
…
a_1, a_2, \ldots
a
1
,
a
2
,
…
satisfies the following conditions.
a
1
=
2
a_1 = 2
a
1
=
2
,
a
2
=
11
a_2 = 11
a
2
=
11
. for all positive integer
n
n
n
,
2
a
n
+
2
=
3
a
n
+
5
(
a
n
2
+
a
n
+
1
2
)
2a_{n+2} =3a_n + \sqrt{5 (a_n^2+a_{n+1}^2)}
2
a
n
+
2
=
3
a
n
+
5
(
a
n
2
+
a
n
+
1
2
)
Prove that
a
n
a_n
a
n
is a rational number for each of positive integer
n
n
n
.
7
1
Hide problems
Move all cards to one deck
Consider
n
n
n
cards with marked numbers
1
1
1
through
n
n
n
. No number have repeted, namely, each number has marked exactly at one card. They are distributed on
n
n
n
boxes so that each box contains exactly one card initially. We want to move all the cards into one box all together according to the following instructions The instruction: Choose an integer
k
(
1
≤
k
≤
n
)
k(1\le k\le n)
k
(
1
≤
k
≤
n
)
, and move a card with number
k
k
k
to the other box such that sum of the number of the card in that box is multiple of
k
k
k
. Find all positive integer
n
n
n
so that there exists a way to gather all the cards in one box.Thanks to @scnwust for correcting wrong translation.
1
1
Hide problems
An acute triangle which is not isosceles
The inscribed circle of an acute triangle
A
B
C
ABC
A
BC
meets the segments
A
B
AB
A
B
and
B
C
BC
BC
at
D
D
D
and
E
E
E
respectively. Let
I
I
I
be the incenter of the triangle
A
B
C
ABC
A
BC
. Prove that the intersection of the line
A
I
AI
A
I
and
D
E
DE
D
E
is on the circle whose diameter is
A
C
AC
A
C
(passing through A, C).