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Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2007 Korea Junior Math Olympiad
2007 Korea Junior Math Olympiad
Part of
Korea Junior Mathematics Olympiad
Subcontests
(8)
1
1
Hide problems
a_ix_i + x_{i+2 }\equiv 0 (mod 5) where a_i sequence, x_i \in N sequence
A sequence
a
1
,
a
2
,
.
.
.
,
a
2007
a_1,a_2,...,a_{2007}
a
1
,
a
2
,
...
,
a
2007
where
a
i
∈
{
2
,
3
}
a_i \in\{2,3\}
a
i
∈
{
2
,
3
}
for
i
=
1
,
2
,
.
.
.
,
2007
i = 1,2,...,2007
i
=
1
,
2
,
...
,
2007
and an integer sequence
x
1
,
x
2
,
.
.
.
,
x
2007
x_1,x_2,...,x_{2007}
x
1
,
x
2
,
...
,
x
2007
satis fies the following:
a
i
x
i
+
x
i
+
2
≡
0
a_ix_i + x_{i+2 }\equiv 0
a
i
x
i
+
x
i
+
2
≡
0
(
m
o
d
5
mod 5
m
o
d
5
) , where the indices are taken modulo
2007
2007
2007
. Prove that
x
1
,
x
2
,
.
.
.
,
x
2007
x_1,x_2,...,x_{2007}
x
1
,
x
2
,
...
,
x
2007
are all multiples of
5
5
5
.
3
1
Hide problems
strings of length 6 composed of three characters a, b, c
Consider the string of length
6
6
6
composed of three characters
a
,
b
,
c
a, b, c
a
,
b
,
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
,
b
a
,
c
c
ab, ba, cc
ab
,
ba
,
cc
). Determine the number of strings that can be reduced to
c
c
c
, the string of length
1
1
1
, by the reducing processes mentioned above.
2
1
Hide problems
(a + b,a^n + b^n) for a,b coprime
If
n
n
n
is a positive integer and
a
,
b
a, b
a
,
b
are relatively prime positive integers, calculate
(
a
+
b
,
a
n
+
b
n
)
(a + b,a^n + b^n)
(
a
+
b
,
a
n
+
b
n
)
.
6
1
Hide problems
f(f(x)) = x, |f(x) - x| \ge 2, f : T\to ={1,2,...,10}
Let
T
=
{
1
,
2
,
.
.
.
,
10
}
T = \{1,2,...,10\}
T
=
{
1
,
2
,
...
,
10
}
. Find the number of bijective functions
f
:
T
→
T
f : T\to T
f
:
T
→
T
that satis es the following for all
x
∈
T
x \in T
x
∈
T
:
f
(
f
(
x
)
)
=
x
f(f(x)) = x
f
(
f
(
x
))
=
x
∣
f
(
x
)
−
x
∣
≥
2
|f(x) - x| \ge 2
∣
f
(
x
)
−
x
∣
≥
2
8
1
Hide problems
prime of the year, n^2 +1 \equiv 0 mod p^{2007}
Prime
p
p
p
is called Prime of the Year if there exists a positive integer
n
n
n
such that
n
2
+
1
≡
0
n^2+ 1 \equiv 0
n
2
+
1
≡
0
(
m
o
d
p
2007
mod p^{2007}
m
o
d
p
2007
). Prove that there are infi nite number of Primes of the Year.
5
1
Hide problems
\frac{a}{c+5b}+\frac{b}{a+5c}+\frac{c}{b+5a}\geq\frac{1}{2} , for a,b,c>0
For all positive real numbers
a
,
b
,
c
.
a, b,c.
a
,
b
,
c
.
Prove the folllowing inequality
a
c
+
5
b
+
b
a
+
5
c
+
c
b
+
5
a
≥
1
2
.
\frac{a}{c+5b}+\frac{b}{a+5c}+\frac{c}{b+5a}\geq\frac{1}{2}.
c
+
5
b
a
+
a
+
5
c
b
+
b
+
5
a
c
≥
2
1
.
7
1
Hide problems
incenter of a triangle is circumcenter to another, 2 isosceles triangles wanted
Let the incircle of
△
A
B
C
\triangle ABC
△
A
BC
meet
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
at
J
,
K
,
L
J,K,L
J
,
K
,
L
. Let
D
(
≠
B
,
J
)
,
E
(
≠
C
,
K
)
,
F
(
≠
A
,
L
)
D(\ne B, J),E(\ne C,K), F(\ne A,L)
D
(
=
B
,
J
)
,
E
(
=
C
,
K
)
,
F
(
=
A
,
L
)
be points on
B
J
,
C
K
,
A
L
BJ,CK,AL
B
J
,
C
K
,
A
L
. If the incenter of
△
A
B
C
\triangle ABC
△
A
BC
is the circumcenter of
△
D
E
F
\triangle DEF
△
D
EF
and
∠
B
A
C
=
∠
D
E
F
\angle BAC = \angle DEF
∠
B
A
C
=
∠
D
EF
, prove that
△
A
B
C
\triangle ABC
△
A
BC
and
△
D
E
F
\triangle DEF
△
D
EF
are isosceles triangles.
4
1
Hide problems
3 perpendicular bisectors and 6 concyclic points given, collinear wanted
Let
P
P
P
be a point inside
△
A
B
C
\triangle ABC
△
A
BC
. Let the perpendicular bisectors of
P
A
,
P
B
,
P
C
PA,PB,PC
P
A
,
PB
,
PC
be
ℓ
1
,
ℓ
2
,
ℓ
3
\ell_1,\ell_2,\ell_3
ℓ
1
,
ℓ
2
,
ℓ
3
. Let
D
=
ℓ
1
∩
ℓ
2
D =\ell_1 \cap \ell_2
D
=
ℓ
1
∩
ℓ
2
,
E
=
ℓ
2
∩
ℓ
3
E=\ell_2 \cap \ell_3
E
=
ℓ
2
∩
ℓ
3
,
F
=
ℓ
3
∩
ℓ
1
F=\ell_3 \cap \ell_1
F
=
ℓ
3
∩
ℓ
1
. If
A
,
B
,
C
,
D
,
E
,
F
A,B,C,D,E,F
A
,
B
,
C
,
D
,
E
,
F
lie on a circle, prove that
C
,
P
,
D
C, P,D
C
,
P
,
D
are collinear.