MathDB
Problems
Contests
National and Regional Contests
Korea Contests
North Korea Team Selection Test
2013 North Korea Team Selection Test
2013 North Korea Team Selection Test
Part of
North Korea Team Selection Test
Subcontests
(6)
6
1
Hide problems
Existence of a solution of a diophantine equation
Show that
x
3
+
x
+
a
2
=
y
2
x^3 + x+ a^2 = y^2
x
3
+
x
+
a
2
=
y
2
has at least one pair of positive integer solution
(
x
,
y
)
(x,y)
(
x
,
y
)
for each positive integer
a
a
a
.
5
1
Hide problems
Incircle of quadrilateral
The incircle
ω
\omega
ω
of a quadrilateral
A
B
C
D
ABCD
A
BC
D
touches
A
B
,
B
C
,
C
D
,
D
A
AB, BC, CD, DA
A
B
,
BC
,
C
D
,
D
A
at
E
,
F
,
G
,
H
E, F, G, H
E
,
F
,
G
,
H
, respectively. Choose an arbitrary point
X
X
X
on the segment
A
C
AC
A
C
inside
ω
\omega
ω
. The segments
X
B
,
X
D
XB, XD
XB
,
X
D
meet
ω
\omega
ω
at
I
,
J
I, J
I
,
J
respectively. Prove that
F
J
,
I
G
,
A
C
FJ, IG, AC
F
J
,
I
G
,
A
C
are concurrent.
4
1
Hide problems
Operation on a 3*3 table
Positive integers 1 to 9 are written in each square of a
3
×
3
3 \times 3
3
×
3
table. Let us define an operation as follows: Take an arbitrary row or column and replace these numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
with either non-negative numbers
a
−
x
,
b
−
x
,
c
+
x
a-x, b-x, c+x
a
−
x
,
b
−
x
,
c
+
x
or
a
+
x
,
b
−
x
,
c
−
x
a+x, b-x, c-x
a
+
x
,
b
−
x
,
c
−
x
, where
x
x
x
is a positive number and can vary in each operation. (1) Does there exist a series of operations such that all 9 numbers turn out to be equal from the following initial arrangement a)? b)?
a
)
1
2
3
4
5
6
7
8
9
)
a) \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array} )
a
)
1
4
7
2
5
8
3
6
9
)
b
)
2
8
5
9
3
4
6
7
1
)
b) \begin{array}{ccc} 2 & 8 & 5 \\ 9 & 3 & 4 \\ 6 & 7 & 1 \end{array} )
b
)
2
9
6
8
3
7
5
4
1
)
(2) Determine the maximum value which all 9 numbers turn out to be equal to after some steps.
3
1
Hide problems
a^n + 2^n divides b^n + c
Find all
a
,
b
,
c
∈
Z
a, b, c \in \mathbb{Z}
a
,
b
,
c
∈
Z
,
c
≥
0
c \ge 0
c
≥
0
such that
a
n
+
2
n
∣
b
n
+
c
a^n + 2^n | b^n + c
a
n
+
2
n
∣
b
n
+
c
for all positive integers
n
n
n
where
2
a
b
2ab
2
ab
is non-square.
2
1
Hide problems
The number of elements is a power of 2
Let
a
1
,
a
2
,
⋯
,
a
k
a_1 , a_2 , \cdots , a_k
a
1
,
a
2
,
⋯
,
a
k
be numbers such that
a
i
∈
{
0
,
1
,
2
,
3
}
(
i
=
1
,
2
,
⋯
,
k
)
a_i \in \{ 0,1,2,3 \} ( i= 1, 2, \cdots ,k)
a
i
∈
{
0
,
1
,
2
,
3
}
(
i
=
1
,
2
,
⋯
,
k
)
. Let
z
=
(
x
k
,
x
k
−
1
,
⋯
,
x
1
)
4
z = ( x_k , x_{k-1} , \cdots , x_1 )_4
z
=
(
x
k
,
x
k
−
1
,
⋯
,
x
1
)
4
be a base 4 expansion of
z
∈
{
0
,
1
,
2
,
⋯
,
4
k
−
1
}
z \in \{ 0, 1, 2, \cdots , 4^k -1 \}
z
∈
{
0
,
1
,
2
,
⋯
,
4
k
−
1
}
. Define
A
A
A
as follows:
A
=
{
z
∣
p
(
z
)
=
z
,
z
=
0
,
1
,
⋯
,
4
k
−
1
}
A = \{ z | p(z)=z, z=0, 1, \cdots ,4^k-1 \}
A
=
{
z
∣
p
(
z
)
=
z
,
z
=
0
,
1
,
⋯
,
4
k
−
1
}
where
p
(
z
)
=
∑
i
=
1
k
a
i
x
i
4
i
−
1
.
p(z) = \sum_{i=1}^{k} a_i x_i 4^{i-1} .
p
(
z
)
=
i
=
1
∑
k
a
i
x
i
4
i
−
1
.
Prove that the number of elements in
X
X
X
is a power of 2.
1
1
Hide problems
Concurrent lines
The incircle of a non-isosceles triangle
A
B
C
ABC
A
BC
with the center
I
I
I
touches the sides
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
at
A
1
,
B
1
,
C
1
A_1 , B_1 , C_1
A
1
,
B
1
,
C
1
respectively. The line
A
I
AI
A
I
meets the circumcircle of
A
B
C
ABC
A
BC
at
A
2
A_2
A
2
. The line
B
1
C
1
B_1 C_1
B
1
C
1
meets the line
B
C
BC
BC
at
A
3
A_3
A
3
and the line
A
2
A
3
A_2 A_3
A
2
A
3
meets the circumcircle of
A
B
C
ABC
A
BC
at
A
4
(
≠
A
2
)
A_4 (\ne A_2 )
A
4
(
=
A
2
)
. Define
B
4
,
C
4
B_4 , C_4
B
4
,
C
4
similarly. Prove that the lines
A
A
4
,
B
B
4
,
C
C
4
AA_4 , BB_4 , CC_4
A
A
4
,
B
B
4
,
C
C
4
are concurrent.