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Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2005 Korea Junior Math Olympiad
2005 Korea Junior Math Olympiad
Part of
Korea Junior Mathematics Olympiad
Subcontests
(8)
3
1
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a_{n+1} =a_n -1 if a_n is even, a_{n+1} =\frac{a_n - 1}{2} if a_n is odd
For a positive integer
K
K
K
, de fine a sequence,
{
a
n
}
\{a_n\}
{
a
n
}
, as following:
a
1
=
K
a_1 = K
a
1
=
K
and
a
n
+
1
=
a
n
−
1
a_{n+1} =a_n -1
a
n
+
1
=
a
n
−
1
if
a
n
a_n
a
n
is even
a
n
+
1
=
a
n
−
1
2
a_{n+1} =\frac{a_n - 1}{2}
a
n
+
1
=
2
a
n
−
1
if
a
n
a_n
a
n
is odd , for all
n
≥
1
n \ge 1
n
≥
1
. Find the smallest value of
K
K
K
, which makes
a
2005
a_{2005}
a
2005
the first term equal to
0
0
0
.
7
1
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KJMO 2005 n-variable inequality
If positive reals
x
1
,
x
2
,
⋯
,
x
n
x_1,x_2,\cdots,x_n
x
1
,
x
2
,
⋯
,
x
n
satisfy
∑
i
=
1
n
x
i
=
1.
\sum_{i=1}^{n}x_i=1.
∑
i
=
1
n
x
i
=
1.
Prove that
∑
i
=
1
n
1
1
+
∑
j
=
1
i
x
j
<
2
3
∑
i
=
1
n
1
x
i
\sum_{i=1}^{n}\frac{1}{1+\sum_{j=1}^{i}x_j}<\sqrt{\frac{2}{3}\sum_{i=1}^{n}\frac{1}{x_i}}
i
=
1
∑
n
1
+
∑
j
=
1
i
x
j
1
<
3
2
i
=
1
∑
n
x
i
1
8
1
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study groups between 6 students
A group of
6
6
6
students decided to make study groups and service activity groups according to the following principle: Each group must have exactly
3
3
3
members. For any pair of students, there are same number of study groups and service activity groups that both of the students are members. Supposing there are at least one group and no three students belong to the same study group and service activity group, prove that the minimum number of groups is
8
8
8
.
6
1
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primes p,q in the form x^2 + 2005y^2, (x, y \in Z)
For two different prime numbers
p
,
q
p, q
p
,
q
, defi ne
S
p
,
q
=
{
p
,
q
,
p
q
}
S_{p,q} = \{p,q,pq\}
S
p
,
q
=
{
p
,
q
,
pq
}
. If two elements in
S
p
,
q
S_{p,q}
S
p
,
q
are numbers in the form of
x
2
+
2005
y
2
,
(
x
,
y
∈
Z
)
x^2 + 2005y^2, (x, y \in Z)
x
2
+
2005
y
2
,
(
x
,
y
∈
Z
)
, prove that all three elements in
S
p
,
q
S_{p,q}
S
p
,
q
are in such form.
4
1
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11 students in a test, for any 2 qestions exactly 6 solved 1 correctly
11
11
11
students take a test. For any two question in a test, there are at least
6
6
6
students who solved exactly one of those two questions. Prove that there are no more than
12
12
12
questions in this test. Showing the equality case is not needed.
1
1
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irreducible fraction with denominator <= 2005, that is closest to 9/25
Find a irreducible fraction with denominator not greater than 2005, that is closest to
9
25
\frac{9}{25}
25
9
but is not
9
25
\frac{9}{25}
25
9
2
1
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length of MN does not depend on P and Q (KJMO 2005 p2)
For triangle
A
B
C
,
P
ABC, P
A
BC
,
P
and
Q
Q
Q
satisfy
∠
B
P
A
+
∠
A
Q
C
=
9
0
o
\angle BPA + \angle AQC = 90^o
∠
BP
A
+
∠
A
QC
=
9
0
o
. It is provided that the vertices of the triangle
B
A
P
BAP
B
A
P
and
A
C
Q
ACQ
A
CQ
are ordered counterclockwise (or clockwise). Let the intersection of the circumcircles of the two triangles be
N
N
N
(
A
≠
N
A \ne N
A
=
N
, however if
A
A
A
is the only intersection
A
=
N
A = N
A
=
N
), and the midpoint of segment
B
C
BC
BC
be
M
M
M
. Show that the length of
M
N
MN
MN
does not depend on
P
P
P
and
Q
Q
Q
.
5
1
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Perpendicular Line Geometry
In
△
A
B
C
\triangle ABC
△
A
BC
, let the bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
hit the circumcircle at
M
M
M
. Let
P
P
P
be the intersection of
C
M
CM
CM
and
A
B
AB
A
B
. Denote by
(
V
,
W
X
,
Y
Z
)
(V,WX,YZ)
(
V
,
W
X
,
Y
Z
)
the intersection of the line passing
V
V
V
perpendicular to
W
X
WX
W
X
with the line
Y
Z
YZ
Y
Z
. Prove that the points
(
P
,
A
M
,
A
C
)
,
(
P
,
A
C
,
A
M
)
,
(
P
,
B
C
,
M
B
)
(P,AM,AC), (P,AC,AM), (P,BC,MB)
(
P
,
A
M
,
A
C
)
,
(
P
,
A
C
,
A
M
)
,
(
P
,
BC
,
MB
)
are collinear.In isosceles triangle
A
P
X
APX
A
PX
with
A
P
=
A
X
AP=AX
A
P
=
A
X
, select a point
M
M
M
on the altitude.
P
M
PM
PM
intersects
A
X
AX
A
X
at
C
C
C
. The circumcircle of
A
C
M
ACM
A
CM
intersects
A
P
AP
A
P
at
B
B
B
. A line passing through
P
P
P
perpendicular to
B
C
BC
BC
intersects
M
B
MB
MB
at
Z
Z
Z
. Show that
X
Z
XZ
XZ
is perpendicular to
A
P
AP
A
P
.