MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2021 Korea Junior Math Olympiad
2021 Korea Junior Math Olympiad
Part of
Korea Junior Mathematics Olympiad
Subcontests
(6)
6
1
Hide problems
Couples and Friends Choosing Numbers
In a meeting of
4042
4042
4042
people, there are
2021
2021
2021
couples, each consisting of two people. Suppose that
A
A
A
and
B
B
B
, in the meeting, are friends when they know each other. For a positive integer
n
n
n
, each people chooses an integer from
−
n
-n
−
n
to
n
n
n
so that the following conditions hold. (Two or more people may choose the same number). [*] Two or less people chose
0
0
0
, and if exactly two people chose
0
0
0
, they are coupled. [*] Two people are either coupled or don't know each other if they chose the same number. [*] Two people are either coupled or know each other if they chose two numbers that sum to
0
0
0
. Determine the least possible value of
n
n
n
for which such number selecting is always possible.
1
1
Hide problems
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
s
,
t
≥
2
s, t \ge 2
s
,
t
≥
2
, prove that
A
(
s
t
,
s
,
t
)
=
A
(
s
(
t
−
1
)
,
s
,
t
)
=
A
(
(
s
−
1
)
t
,
s
,
t
)
.
A(st, s, t) = A(s(t-1), s, t) = A((s-1)t, s, t).
A
(
s
t
,
s
,
t
)
=
A
(
s
(
t
−
1
)
,
s
,
t
)
=
A
((
s
−
1
)
t
,
s
,
t
)
.
4
1
Hide problems
Collinear iff Concyclic
In an acute triangle
A
B
C
ABC
A
BC
with
A
B
‾
<
A
C
‾
\overline{AB} < \overline{AC}
A
B
<
A
C
, angle bisector of
A
A
A
and perpendicular bisector of
B
C
‾
\overline{BC}
BC
intersect at
D
D
D
. Let
P
P
P
be an interior point of triangle
A
B
C
ABC
A
BC
. Line
C
P
CP
CP
meets the circumcircle of triangle
A
B
P
ABP
A
BP
again at
K
K
K
. Prove that
B
,
D
,
K
B, D, K
B
,
D
,
K
are collinear if and only if
A
D
AD
A
D
and
B
C
BC
BC
meet on the circumcircle of triangle
A
P
C
APC
A
PC
.
5
1
Hide problems
Korean Functional Equation With Square
Determine all functions
f
:
R
→
R
f \colon \mathbb{R} \to \mathbb{R}
f
:
R
→
R
satisfying
f
(
f
(
x
+
y
)
−
f
(
x
−
y
)
)
=
y
2
f
(
x
)
f(f(x+y)-f(x-y))=y^2f(x)
f
(
f
(
x
+
y
)
−
f
(
x
−
y
))
=
y
2
f
(
x
)
for all
x
,
y
∈
R
x, y \in \mathbb{R}
x
,
y
∈
R
.
2
1
Hide problems
Modulo Sequence of 2021
Let
{
a
n
}
\{a_n\}
{
a
n
}
be a sequence of integers satisfying the following conditions. [*]
a
1
=
202
1
2021
a_1=2021^{2021}
a
1
=
202
1
2021
[*]
0
≤
a
k
<
k
0 \le a_k < k
0
≤
a
k
<
k
for all integers
k
≥
2
k \ge 2
k
≥
2
[*]
a
1
−
a
2
+
a
3
−
a
4
+
⋯
+
(
−
1
)
k
+
1
a
k
a_1-a_2+a_3-a_4+ \cdots + (-1)^{k+1}a_k
a
1
−
a
2
+
a
3
−
a
4
+
⋯
+
(
−
1
)
k
+
1
a
k
is multiple of
k
k
k
for all positive integers
k
k
k
. Determine the
202
1
2022
2021^{2022}
202
1
2022
th term of the sequence
{
a
n
}
\{a_n\}
{
a
n
}
.
3
1
Hide problems
Parallel Line Tangent to a Circle
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral with circumcircle
Ω
\Omega
Ω
and let diagonals
A
C
AC
A
C
and
B
D
BD
B
D
intersect at
X
X
X
. Suppose that
A
E
F
B
AEFB
A
EFB
is inscribed in a circumcircle of triangle
A
B
X
ABX
A
BX
such that
E
F
EF
EF
and
A
B
AB
A
B
are parallel.
F
X
FX
FX
meets the circumcircle of triangle
C
D
X
CDX
C
D
X
again at
G
G
G
. Let
E
X
EX
EX
meets
A
B
AB
A
B
at
P
P
P
, and
X
G
XG
XG
meets
C
D
CD
C
D
at
Q
Q
Q
. Denote by
S
S
S
the intersection of the perpendicular bisector of
E
G
‾
\overline{EG}
EG
and
Ω
\Omega
Ω
such that
S
S
S
is closer to
A
A
A
than
B
B
B
. Prove that line through
S
S
S
parallel to
P
Q
PQ
PQ
is tangent to
Ω
\Omega
Ω
.