MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2018 Korea Junior Math Olympiad
2018 Korea Junior Math Olympiad
Part of
Korea Junior Mathematics Olympiad
Subcontests
(8)
8
1
Hide problems
Evading function
For every set
S
S
S
with
n
(
≥
3
)
n(\ge3)
n
(
≥
3
)
distinct integers, show that there exists a function
f
:
{
1
,
2
,
…
,
n
}
→
S
f:\{1,2,\dots,n\}\rightarrow S
f
:
{
1
,
2
,
…
,
n
}
→
S
satisfying the following two conditions.(i)
{
f
(
1
)
,
f
(
2
)
,
…
,
f
(
n
)
}
=
S
\{ f(1),f(2),\dots,f(n)\} = S
{
f
(
1
)
,
f
(
2
)
,
…
,
f
(
n
)}
=
S
(ii)
2
f
(
j
)
≠
f
(
i
)
+
f
(
k
)
2f(j)\neq f(i)+f(k)
2
f
(
j
)
=
f
(
i
)
+
f
(
k
)
for all
1
≤
i
<
j
<
k
≤
n
1\le i<j<k\le n
1
≤
i
<
j
<
k
≤
n
.
7
1
Hide problems
diophantine eqn
Find all integer pair
(
m
,
n
)
(m,n)
(
m
,
n
)
such that
7
m
=
5
n
+
24
7^m=5^n+24
7
m
=
5
n
+
24
.
6
1
Hide problems
Minimize the sum of difference squares
Let there be a figure with
9
9
9
disks and
11
11
11
edges, as shown below.We will write a real number in each and every disk. Then, for each edge, we will write the square of the difference between the two real numbers written in the two disks that the edge connects. We must write
0
0
0
in disk
A
A
A
, and
1
1
1
in disk
I
I
I
. Find the minimum sum of all real numbers written in
11
11
11
edges.
5
1
Hide problems
Reflection of circumcenter
Let there be an acute scalene triangle
A
B
C
ABC
A
BC
with circumcenter
O
O
O
. Denote
D
,
E
D,E
D
,
E
be the reflection of
O
O
O
with respect to
A
B
,
A
C
AB,AC
A
B
,
A
C
, respectively. The circumcircle of
A
D
E
ADE
A
D
E
meets
A
B
AB
A
B
,
A
C
AC
A
C
, the circumcircle of
A
B
C
ABC
A
BC
at points
K
,
L
,
M
K,L,M
K
,
L
,
M
, respectively, and they are all distinct from
A
A
A
. Prove that the lines
B
C
,
K
L
,
A
M
BC,KL,AM
BC
,
K
L
,
A
M
are concurrent.
4
1
Hide problems
Counting solutions of diophantine equation
For a positive integer
n
n
n
, denote
p
(
n
)
p(n)
p
(
n
)
to be the number of nonnegative integer tuples
(
x
,
y
,
z
,
w
)
(x,y,z,w)
(
x
,
y
,
z
,
w
)
such that
x
+
2
y
+
2
z
+
3
w
=
n
x+2y+2z+3w=n
x
+
2
y
+
2
z
+
3
w
=
n
. Also, denote
q
(
n
)
q(n)
q
(
n
)
to be the number of nonnegative integer tuples
(
a
,
b
,
c
,
d
)
(a,b,c,d)
(
a
,
b
,
c
,
d
)
such that(i)
a
+
b
+
c
+
d
=
n
a+b+c+d=n
a
+
b
+
c
+
d
=
n
(ii)
a
≥
b
≥
d
a \ge b \ge d
a
≥
b
≥
d
(iii)
a
≥
c
≥
d
a \ge c \ge d
a
≥
c
≥
d
Prove that for all
n
n
n
,
p
(
n
)
=
q
(
n
)
p(n) = q(n)
p
(
n
)
=
q
(
n
)
.
3
1
Hide problems
incenters of half the triangle
Let there be a scalene triangle
A
B
C
ABC
A
BC
, and denote
M
M
M
by the midpoint of
B
C
BC
BC
. The perpendicular bisector of
B
C
BC
BC
meets the circumcircle of
A
B
C
ABC
A
BC
at point
P
P
P
, on the same side with
A
A
A
with respect to
B
C
BC
BC
. Let the incenters of
A
B
M
ABM
A
BM
and
A
M
C
AMC
A
MC
be
I
,
J
I,J
I
,
J
, respectively. Let
∠
B
A
C
=
α
\angle BAC=\alpha
∠
B
A
C
=
α
,
∠
A
B
C
=
β
\angle ABC=\beta
∠
A
BC
=
β
,
∠
B
C
A
=
γ
\angle BCA=\gamma
∠
BC
A
=
γ
. Find
∠
I
P
J
\angle IPJ
∠
I
P
J
.
2
1
Hide problems
sum of squares of divisors
Find all positive integer
N
N
N
which has not less than
4
4
4
positive divisors, such that the sum of squares of the
4
4
4
smallest positive divisors of
N
N
N
is equal to
N
N
N
.
1
1
Hide problems
Quadratic function
Let
f
f
f
be a quadratic function which satisfies the following condition. Find the value of
f
(
8
)
−
f
(
2
)
f
(
2
)
−
f
(
1
)
\frac{f(8)-f(2)}{f(2)-f(1)}
f
(
2
)
−
f
(
1
)
f
(
8
)
−
f
(
2
)
.For two distinct real numbers
a
,
b
a,b
a
,
b
, if
f
(
a
)
=
f
(
b
)
f(a)=f(b)
f
(
a
)
=
f
(
b
)
, then
f
(
a
2
−
6
b
−
1
)
=
f
(
b
2
+
8
)
f(a^2-6b-1)=f(b^2+8)
f
(
a
2
−
6
b
−
1
)
=
f
(
b
2
+
8
)
.