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Problems
Contests
National and Regional Contests
Korea Contests
Korea National Olympiad
2007 Korea National Olympiad
2007 Korea National Olympiad
Part of
Korea National Olympiad
Subcontests
(4)
1
2
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string of length 6 composed of three characters
Consider the string of length
6
6
6
composed of three characters
a
a
a
,
b
b
b
,
c
c
c
. For each string, if two
a
a
a
s are next to each other, or two
b
b
b
s are next to each other, then replace
a
a
aa
aa
by
b
b
b
, and replace
b
b
bb
bb
by
a
a
a
. Also, if
a
a
a
and
b
b
b
are next to each other, or two
c
c
c
s are next to each other, remove all two of them (i.e. delete
a
b
ab
ab
,
b
a
ba
ba
,
c
c
cc
cc
). Determine the number of strings that can be reduced to
c
c
c
, the string of length 1, by the reducing processes mentioned above.
what is the value of positive constant k?
For all positive reals
a
a
a
,
b
b
b
, and
c
c
c
, what is the value of positive constant
k
k
k
satisfies the following inequality? \frac{a}{c\plus{}kb}\plus{}\frac{b}{a\plus{}kc}\plus{}\frac{c}{b\plus{}ka}\geq\frac{1}{2007} .
2
2
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convex quadrilateral
A
1
B
1
B
2
A
2
A_{1}B_{1}B_{2}A_{2}
A
1
B
1
B
2
A
2
is a convex quadrilateral, and
A
1
B
1
≠
A
2
B
2
A_{1}B_{1}\neq A_{2}B_{2}
A
1
B
1
=
A
2
B
2
. Show that there exists a point
M
M
M
such that \frac{A_{1}B_{1}}{A_{2}B_{2}}\equal{}\frac{MA_{1}}{MA_{2}}\equal{}\frac{MB_{1}}{MB_{2}}
triangle is not isosceles
A
B
C
ABC
A
BC
is a triangle which is not isosceles. Let the circumcenter and orthocenter of
A
B
C
ABC
A
BC
be
O
O
O
,
H
H
H
, respectively, and the altitudes of
A
B
C
ABC
A
BC
be
A
D
AD
A
D
,
B
C
BC
BC
,
C
F
CF
CF
. Let
K
≠
A
K\neq A
K
=
A
be the intersection of
A
D
AD
A
D
and circumcircle of triangle
A
B
C
ABC
A
BC
,
L
L
L
be the intersection of
O
K
OK
O
K
and
B
C
BC
BC
,
M
M
M
be the midpoint of
B
C
BC
BC
,
P
P
P
be the intersection of
A
M
AM
A
M
and the line that passes
L
L
L
and perpendicular to
B
C
BC
BC
,
Q
Q
Q
be the intersection of
A
D
AD
A
D
and the line that passes
P
P
P
and parallel to
M
H
MH
M
H
,
R
R
R
be the intersection of line
E
Q
EQ
EQ
and
A
B
AB
A
B
,
S
S
S
be the intersection of
F
D
FD
F
D
and
B
E
BE
BE
. If OL \equal{} KL, then prove that two lines
O
H
OH
O
H
and
R
S
RS
RS
are orthogonal.
3
2
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on the set of all positive integers whose all digits are 1,2
Let
S
S
S
be the set of all positive integers whose all digits are
1
1
1
or
2
2
2
. Denote
T
n
T_{n}
T
n
as the set of all integers which is divisible by
n
n
n
, then find all positive integers
n
n
n
such that
S
∩
T
n
S\cap T_{n}
S
∩
T
n
is an infinite set.
chess board has size is 2007x2007
In each
200
7
2
2007^{2}
200
7
2
unit squares on chess board whose size is
2007
×
2007
2007\times 2007
2007
×
2007
, there lies one coin each square such that their "heads" face upward. Consider the process that flips four consecutive coins on the same row, or flips four consecutive coins on the same column. Doing this process finite times, we want to make the "tails" of all of coins face upward, except one that lies in the
i
i
i
th row and
j
j
j
th column. Show that this is possible if and only if both of
i
i
i
and
j
j
j
are divisible by
4
4
4
.
4
2
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two real sequence
Two real sequence
{
x
n
}
\{x_{n}\}
{
x
n
}
and
{
y
n
}
\{y_{n}\}
{
y
n
}
satisfies following recurrence formula; x_{0}\equal{} 1, y_{0}\equal{} 2007 x_{n\plus{}1}\equal{} x_{n}\minus{}(x_{n}y_{n}\plus{}x_{n\plus{}1}y_{n\plus{}1}\minus{}2)(y_{n}\plus{}y_{n\plus{}1}), y_{n\plus{}1}\equal{} y_{n}\minus{}(x_{n}y_{n}\plus{}x_{n\plus{}1}y_{n\plus{}1}\minus{}2)(x_{n}\plus{}x_{n\plus{}1}) Then show that for all nonnegative integer
n
n
n
,
x
n
2
≤
2007
{x_{n}}^{2}\leq 2007
x
n
2
≤
2007
.
(2007 KMO #8) Product of primes less than n
For all positive integer
n
≥
2
n\geq 2
n
≥
2
, prove that product of all prime numbers less or equal than
n
n
n
is smaller than
4
n
4^{n}
4
n
.