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Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2009 Korea Junior Math Olympiad
2009 Korea Junior Math Olympiad
Part of
Korea Junior Mathematics Olympiad
Subcontests
(8)
7
1
Hide problems
9 students from 3 countries in a circular table
There are
3
3
3
students from Korea, China, and Japan, so total of
9
9
9
students are present. How many ways are there to make them sit down in a circular table, with equally spaced and equal chairs, such that the students from the same country do not sit next to each other? If array
A
A
A
can become array
B
B
B
by rotation, these two arrays are considered equal.
5
1
Hide problems
equal segments from KJMO 2009
Acute triangle
△
A
B
C
\triangle ABC
△
A
BC
satis es
A
B
<
A
C
AB < AC
A
B
<
A
C
. Let the circumcircle of this triangle be
O
O
O
, and the midpoint of
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
be
D
,
E
,
F
D,E,F
D
,
E
,
F
. Let
P
P
P
be the intersection of the circle with
A
B
AB
A
B
as its diameter and line
D
F
DF
D
F
, which is in the same side of
C
C
C
with respect to
A
B
AB
A
B
. Let
Q
Q
Q
be the intersection of the circle with
A
C
AC
A
C
as its diameter and the line
D
E
DE
D
E
, which is in the same side of
B
B
B
with respect to
A
C
AC
A
C
. Let
P
Q
∩
B
C
=
R
PQ \cap BC = R
PQ
∩
BC
=
R
, and let the line passing through
R
R
R
and perpendicular to
B
C
BC
BC
meet
A
O
AO
A
O
at
X
X
X
. Prove that
A
X
=
X
R
AX = XR
A
X
=
XR
.
2
1
Hide problems
concurrent lines constructed by reflections
In an acute triangle
△
A
B
C
\triangle ABC
△
A
BC
, let
A
′
,
B
′
,
C
′
A',B',C'
A
′
,
B
′
,
C
′
be the reflection of
A
,
B
,
C
A,B,C
A
,
B
,
C
with respect to
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
. Let
D
=
B
′
C
∩
B
C
′
D = B'C \cap BC'
D
=
B
′
C
∩
B
C
′
,
E
=
C
A
′
∩
C
′
A
E = CA' \cap C'A
E
=
C
A
′
∩
C
′
A
,
F
=
A
′
B
∩
A
B
′
F = A'B \cap AB'
F
=
A
′
B
∩
A
B
′
. Prove that
A
D
,
B
E
,
C
F
AD,BE,CF
A
D
,
BE
,
CF
are concurrent
1
1
Hide problems
find abc when a\(b+8), a\(b^2 - 1), c\(b^2 - 1), b + c = a^2-1, primes a,b,c
For primes
a
,
b
,
c
a, b,c
a
,
b
,
c
that satis fy the following, calculate
a
b
c
abc
ab
c
.
∙
\bullet
∙
b
+
8
b + 8
b
+
8
is a multiple of
a
a
a
,
∙
\bullet
∙
b
2
−
1
b^2 - 1
b
2
−
1
is a multiple of
a
a
a
and
c
c
c
∙
\bullet
∙
b
+
c
=
a
2
−
1
b + c = a^2 - 1
b
+
c
=
a
2
−
1
.
6
1
Hide problems
1<\frac{b}{ab+b+1}+\frac{c}{bc+c+1}+\frac{d}{cd+d+1}+\frac{a}{da+a+1}<2
If positive reals
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
satisfy
a
b
c
d
=
1.
abcd = 1.
ab
c
d
=
1.
Prove the following inequality
1
<
b
a
b
+
b
+
1
+
c
b
c
+
c
+
1
+
d
c
d
+
d
+
1
+
a
d
a
+
a
+
1
<
2.
1<\frac{b}{ab+b+1}+\frac{c}{bc+c+1}+\frac{d}{cd+d+1}+\frac{a}{da+a+1}<2.
1
<
ab
+
b
+
1
b
+
b
c
+
c
+
1
c
+
c
d
+
d
+
1
d
+
d
a
+
a
+
1
a
<
2.
3
1
Hide problems
\frac{x^2}{x+y}+\frac{y^2}{1-x}+\frac{(1-x-y)^2}{1-y}\geq\frac{1}{2} if 0<x,y<1
For two arbitrary reals
x
,
y
x, y
x
,
y
which are larger than
0
0
0
and less than
1.
1.
1.
Prove that
x
2
x
+
y
+
y
2
1
−
x
+
(
1
−
x
−
y
)
2
1
−
y
≥
1
2
.
\frac{x^2}{x+y}+\frac{y^2}{1-x}+\frac{(1-x-y)^2}{1-y}\geq\frac{1}{2}.
x
+
y
x
2
+
1
−
x
y
2
+
1
−
y
(
1
−
x
−
y
)
2
≥
2
1
.
8
1
Hide problems
Prime Powers
Let a, b, c, d, and e be positive integers. Are there any solutions to
a
2
+
b
3
+
c
5
+
d
7
=
e
11
a^2+b^3+c^5+d^7=e^{11}
a
2
+
b
3
+
c
5
+
d
7
=
e
11
?
4
1
Hide problems
Students in Clubs
There are
n
n
n
clubs composed of
4
4
4
students out of all
9
9
9
students. For two arbitrary clubs, there are no more than
2
2
2
students who are a member of both clubs. Prove that
n
≤
18
n\le 18
n
≤
18
. Translator’s Note. We can prove
n
≤
12
n\le 12
n
≤
12
, and we can prove that the bound is tight.(Credits to rkm0959 for translation and document)