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Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2013 Korea Junior Math Olympiad
2013 Korea Junior Math Olympiad
Part of
Korea Junior Mathematics Olympiad
Subcontests
(7)
3
1
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b_n=\frac{1}{a_{2n+1}}\Sigma_{i=1}^{4n-2}a_i is positive integer
{
a
n
}
\{a_n\}
{
a
n
}
is a positive integer sequence such that
a
i
+
2
=
a
i
+
1
+
a
i
a_{i+2} = a_{i+1} +a_i
a
i
+
2
=
a
i
+
1
+
a
i
(for all
i
≥
1
i \ge 1
i
≥
1
). For positive integer
n
n
n
, de fine as
b
n
=
1
a
2
n
+
1
Σ
i
=
1
4
n
−
2
a
i
b_n=\frac{1}{a_{2n+1}}\Sigma_{i=1}^{4n-2}a_i
b
n
=
a
2
n
+
1
1
Σ
i
=
1
4
n
−
2
a
i
Prove that
b
n
b_n
b
n
is positive integer.
2
1
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tangent perpendicular to line, starting with a pentagon incribed in a circle
A pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
is inscribed in a circle
O
O
O
, and satis es
A
B
=
B
C
,
A
E
=
D
E
AB = BC , AE = DE
A
B
=
BC
,
A
E
=
D
E
. The circle that is tangent to
D
E
DE
D
E
at
E
E
E
and passing
A
A
A
hits
E
C
EC
EC
at
F
F
F
and
B
F
BF
BF
at
G
(
≠
F
)
G (\ne F)
G
(
=
F
)
. Let
D
G
∩
O
=
H
(
≠
D
)
DG\cap O = H (\ne D)
D
G
∩
O
=
H
(
=
D
)
. Prove that the tangent to
O
O
O
at
E
E
E
is perpendicular to
H
A
HA
H
A
.
4
1
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exists prime p so that min n such p|2^n-1 is 3^{2013}
Prove that there exists a prime number
p
p
p
such that the minimum positive integer
n
n
n
such that
p
∣
2
n
−
1
p|2^n -1
p
∣
2
n
−
1
is
3
2013
3^{2013}
3
2013
.
5
1
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intersection of two lines lies on the circumcircle
In an acute triangle
△
A
B
C
,
∠
A
>
∠
B
\triangle ABC, \angle A > \angle B
△
A
BC
,
∠
A
>
∠
B
. Let the midpoint of
A
B
AB
A
B
be
D
D
D
, and let the foot of the perpendicular from
A
A
A
to
B
C
BC
BC
be
E
E
E
, and
B
B
B
from
C
A
CA
C
A
be
F
F
F
. Let the circumcenter of
△
D
E
F
\triangle DEF
△
D
EF
be
O
O
O
. A point
J
J
J
on segment
B
E
BE
BE
satisfi es
∠
O
D
C
=
∠
E
A
J
\angle ODC = \angle EAJ
∠
O
D
C
=
∠
E
A
J
. Prove that
A
J
∩
D
C
AJ \cap DC
A
J
∩
D
C
lies on the circumcircle of
△
B
D
E
\triangle BDE
△
B
D
E
.
8
1
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one 2013-gon constructed by diagonals of a regular 2013-gon,
Drawing all diagonals in a regular
2013
2013
2013
-gon, the regular
2013
2013
2013
-gon is divided into non-overlapping polygons. Prove that there exist exactly one
2013
2013
2013
-gon out of all such polygons.
7
1
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Function with asymptotic irrational slope
Let
f
:
N
⟶
N
f:\mathbb{N} \longrightarrow \mathbb{N}
f
:
N
⟶
N
be such that for every positive integer
n
n
n
, followings are satisfied. i.
f
(
n
+
1
)
>
f
(
n
)
f(n+1) > f(n)
f
(
n
+
1
)
>
f
(
n
)
ii.
f
(
f
(
n
)
)
=
2
n
+
2
f(f(n)) = 2n+2
f
(
f
(
n
))
=
2
n
+
2
Find the value of
f
(
2013
)
f(2013)
f
(
2013
)
. (Here,
N
\mathbb{N}
N
is the set of all positive integers.)
1
1
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Comparison between numbers
Compare the magnitude of the following three numbers.
25
3
3
,
1148
135
3
,
25
3
3
+
6
5
3
\sqrt[3]{\frac{25}{3}} ,\, \sqrt[3]{\frac{1148}{135}} ,\, \frac{\sqrt[3]{25}}{3} + \sqrt[3]{\frac{6}{5}}
3
3
25
,
3
135
1148
,
3
3
25
+
3
5
6