MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2014 Korea Junior Math Olympiad
2014 Korea Junior Math Olympiad
Part of
Korea Junior Mathematics Olympiad
Subcontests
(8)
3
1
Hide problems
how many ways are there to move n times following the arrows
Find the number of
n
n
n
-movement on the following graph, starting from
S
S
S
. https://cdn.artofproblemsolving.com/attachments/2/0/4a23c83c7f5405575acbe6d09f202d87341337.png
2
1
Hide problems
s/x_1+t/x_2+s/x_3+t/x_4+...+s/x_{2n-1}+t/x_{2n} > 2n^2/(n+1)
Let there be
2
n
2n
2
n
positive reals
a
1
,
a
2
,
.
.
.
,
a
2
n
a_1,a_2,...,a_{2n}
a
1
,
a
2
,
...
,
a
2
n
. Let
s
=
a
1
+
a
3
+
.
.
.
+
a
2
n
−
1
s = a_1 + a_3 +...+ a_{2n-1}
s
=
a
1
+
a
3
+
...
+
a
2
n
−
1
,
t
=
a
2
+
a
4
+
.
.
.
+
a
2
n
t = a_2 + a_4 + ... + a_{2n}
t
=
a
2
+
a
4
+
...
+
a
2
n
, and
x
k
=
a
k
+
a
k
+
1
+
.
.
.
+
a
k
+
n
−
1
x_k = a_k + a_{k+1} + ... + a_{k+n-1}
x
k
=
a
k
+
a
k
+
1
+
...
+
a
k
+
n
−
1
(indices are taken modulo
2
n
2n
2
n
). Prove that
s
x
1
+
t
x
2
+
s
x
3
+
t
x
4
+
.
.
.
+
s
x
2
n
−
1
+
t
x
2
n
>
2
n
2
n
+
1
\frac{s}{x_1}+\frac{t}{x_2}+\frac{s}{x_3}+\frac{t}{x_4}+...+\frac{s}{x_{2n-1}}+\frac{t}{x_{2n}}>\frac{2n^2}{n+1}
x
1
s
+
x
2
t
+
x
3
s
+
x
4
t
+
...
+
x
2
n
−
1
s
+
x
2
n
t
>
n
+
1
2
n
2
8
1
Hide problems
n students in m clubs
Let there be
n
n
n
students and
m
m
m
clubs. The students joined the clubs so that the following is true: - For all students
x
x
x
, you can choose some clubs such that
x
x
x
is the only student who joined all of the chosen clubs. Let the number of clubs each student joined be
a
1
,
a
2
,
.
.
.
,
a
m
a_1,a_2,...,a_m
a
1
,
a
2
,
...
,
a
m
. Prove that
a
1
!
(
m
−
a
1
)
!
+
a
2
!
(
m
−
a
2
)
!
+
.
.
.
+
a
n
!
(
m
−
a
n
)
!
≤
m
!
a_1!(m - a_1)! + a_2!(m - a_2)! + ... + a_n!(m -a_n)! \le m!
a
1
!
(
m
−
a
1
)!
+
a
2
!
(
m
−
a
2
)!
+
...
+
a
n
!
(
m
−
a
n
)!
≤
m
!
5
1
Hide problems
(x^2y + x) is a multiple of (xy^2 + 7) , find positive integers x,y
For positive integers
x
,
y
x,y
x
,
y
, find all pairs
(
x
,
y
)
(x,y)
(
x
,
y
)
such that
x
2
y
+
x
x^2y + x
x
2
y
+
x
is a multiple of
x
y
2
+
7
xy^2 + 7
x
y
2
+
7
.
4
1
Hide problems
gcd(a,b,c) = 1 , exists a so that gcd(p,q+ar) = 1
Positive integers
p
,
q
,
r
p, q, r
p
,
q
,
r
satisfy
g
c
d
(
a
,
b
,
c
)
=
1
gcd(a,b,c) = 1
g
c
d
(
a
,
b
,
c
)
=
1
. Prove that there exists an integer
a
a
a
such that
g
c
d
(
p
,
q
+
a
r
)
=
1
gcd(p,q+ar) = 1
g
c
d
(
p
,
q
+
a
r
)
=
1
.
6
1
Hide problems
max of x^p+y^p+z^p when (x-1)^2 +(y-1)^2+(z-1)^2 = 27, p= 1+1/2+...+1/2^5
Let
p
=
1
+
1
2
+
1
2
2
+
1
2
3
+
1
2
4
+
1
2
5
.
p = 1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}+\frac{1}{2^5}.
p
=
1
+
2
1
+
2
2
1
+
2
3
1
+
2
4
1
+
2
5
1
.
For nonnegative reals
x
,
y
,
z
x, y,z
x
,
y
,
z
satisfying
(
x
−
1
)
2
+
(
y
−
1
)
2
+
(
z
−
1
)
2
=
27
,
(x-1)^2 + (y-1)^2 + (z-1)^2 = 27,
(
x
−
1
)
2
+
(
y
−
1
)
2
+
(
z
−
1
)
2
=
27
,
find the maximum value of
x
p
+
y
p
+
z
p
.
x^p + y^p + z^p.
x
p
+
y
p
+
z
p
.
1
1
Hide problems
concyclic Points
Given
△
A
B
C
\triangle ABC
△
A
BC
with incenter
I
I
I
. Line
A
I
AI
A
I
meets
B
C
BC
BC
at
D
D
D
. The incenter of
△
A
B
D
,
△
A
D
C
\triangle ABD, \triangle ADC
△
A
B
D
,
△
A
D
C
are
E
,
F
E,F
E
,
F
, respectively. Line
D
E
DE
D
E
meets the circumcircle of
△
B
C
E
\triangle BCE
△
BCE
at
P
(
≠
E
)
P(\neq E)
P
(
=
E
)
and line
D
F
DF
D
F
meets the circumcircle of
△
B
C
F
\triangle BCF
△
BCF
at
Q
(
≠
F
)
Q(\neq F)
Q
(
=
F
)
. Show that the midpoint of
B
C
BC
BC
lies on the circumcircle of
△
D
P
Q
\triangle DPQ
△
D
PQ
.
7
1
Hide problems
Menelaus and Length Bashing
In a parallelogram
□
A
B
C
D
\Box ABCD
□
A
BC
D
(
A
B
<
B
C
)
(AB < BC)
(
A
B
<
BC
)
The incircle of
△
A
B
C
\triangle ABC
△
A
BC
meets
B
C
‾
\overline {BC}
BC
and
C
A
‾
\overline {CA}
C
A
at
P
,
Q
P, Q
P
,
Q
. The incircle of
△
A
C
D
\triangle ACD
△
A
C
D
and
C
D
‾
\overline {CD}
C
D
meets at
R
R
R
. Let
S
S
S
=
P
Q
PQ
PQ
∩
\cap
∩
A
D
AD
A
D
U
U
U
=
A
R
AR
A
R
∩
\cap
∩
C
S
CS
CS
T
T
T
, a point on
B
C
‾
\overline {BC}
BC
such that
A
B
‾
=
B
T
‾
\overline {AB} = \overline {BT}
A
B
=
BT
Prove that
A
T
,
B
U
,
P
Q
AT, BU, PQ
A
T
,
B
U
,
PQ
are concurrent