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Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2011 Korea Junior Math Olympiad
2011 Korea Junior Math Olympiad
Part of
Korea Junior Mathematics Olympiad
Subcontests
(8)
8
1
Hide problems
n students each having r positive integers, nr integers all different, classes
There are
n
n
n
students each having
r
r
r
positive integers. Their
n
r
nr
n
r
positive integers are all different. Prove that we can divide the students into
k
k
k
classes satisfying the following conditions: (a)
k
≤
4
r
k \le 4r
k
≤
4
r
(b) If a student
A
A
A
has the number
m
m
m
, then the student
B
B
B
in the same class can't have a number
ℓ
\ell
ℓ
such that
(
m
−
1
)
!
<
ℓ
<
(
m
+
1
)
!
+
1
(m - 1)! < \ell < (m + 1)! + 1
(
m
−
1
)!
<
ℓ
<
(
m
+
1
)!
+
1
6
1
Hide problems
f(n) is the sum of \phi (a)\phi (b) for all (a, b) \in S_n
For a positive integer
n
n
n
, define the set
S
n
S_n
S
n
as
S
n
=
{
(
a
,
b
)
∣
a
,
b
∈
N
,
l
c
m
[
a
,
b
]
=
n
}
S_n =\{(a, b)|a, b \in N, lcm[a, b] = n\}
S
n
=
{(
a
,
b
)
∣
a
,
b
∈
N
,
l
c
m
[
a
,
b
]
=
n
}
. Let
f
(
n
)
f(n)
f
(
n
)
be the sum of
ϕ
(
a
)
ϕ
(
b
)
\phi (a)\phi (b)
ϕ
(
a
)
ϕ
(
b
)
for all
(
a
,
b
)
∈
S
n
(a, b) \in S_n
(
a
,
b
)
∈
S
n
. If a prime
p
p
p
relatively prime to
n
n
n
is a divisor of
f
(
n
)
f(n)
f
(
n
)
, prove that there exists a prime
q
∣
n
q|n
q
∣
n
such that
p
∣
q
2
−
1
p|q^2 - 1
p
∣
q
2
−
1
.
4
1
Hide problems
number of sets with 2n + 1 points in coordinate plane
For a positive integer
n
n
n
, (
n
≥
2
n\ge 2
n
≥
2
), find the number of sets with
2
n
+
1
2n + 1
2
n
+
1
points
P
0
,
P
1
,
.
.
.
,
P
2
n
P_0, P_1,..., P_{2n}
P
0
,
P
1
,
...
,
P
2
n
in the coordinate plane satisfying the following as its elements: -
P
0
=
(
0
,
0
)
,
P
2
n
=
(
n
,
n
)
P_0 = (0, 0),P_{2n}= (n, n)
P
0
=
(
0
,
0
)
,
P
2
n
=
(
n
,
n
)
- For all
i
=
1
,
2
,
.
.
.
,
2
n
−
1
i = 1,2,..., 2n - 1
i
=
1
,
2
,
...
,
2
n
−
1
, line
P
i
P
i
+
1
P_iP_{i+1}
P
i
P
i
+
1
is parallel to
x
x
x
-axis or
y
y
y
-axis and its length is
1
1
1
. - Out of
2
n
2n
2
n
lines
P
0
P
1
,
P
1
P
2
,
.
.
.
,
P
2
n
−
1
P
2
n
P_0P_1, P_1P_2,..., P_{2n-1}P_{2n}
P
0
P
1
,
P
1
P
2
,
...
,
P
2
n
−
1
P
2
n
, there are exactly
4
4
4
lines that are enclosed in the domain
y
≤
x
y \le x
y
≤
x
.
3
1
Hide problems
x,y coprime, x + 3y^2 perfect square, x^2 + 9y^4 not a perfect square
Let
x
,
y
x, y
x
,
y
be positive integers such that
g
c
d
(
x
,
y
)
=
1
gcd(x, y) = 1
g
c
d
(
x
,
y
)
=
1
and
x
+
3
y
2
x + 3y^2
x
+
3
y
2
is a perfect square. Prove that
x
2
+
9
y
4
x^2 + 9y^4
x
2
+
9
y
4
can't be a perfect square.
2
1
Hide problems
equal segments iff 3 points are collinear, 3 circles related
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral inscirbed in circle
O
O
O
. Let the tangent to
O
O
O
at
A
A
A
meet
B
C
BC
BC
at
S
S
S
, and the tangent to
O
O
O
at
B
B
B
meet
C
D
CD
C
D
at
T
T
T
. Circle with
S
S
S
as its center and passing
A
A
A
meets
B
C
BC
BC
at
E
E
E
, and
A
E
AE
A
E
meets
O
O
O
again at
F
(
≠
A
)
F(\ne A)
F
(
=
A
)
. The circle with
T
T
T
as its center and passing
B
B
B
meets
C
D
CD
C
D
at
K
K
K
. Let
P
=
B
K
∩
A
C
P = BK \cap AC
P
=
B
K
∩
A
C
. Prove that
P
,
F
,
D
P,F,D
P
,
F
,
D
are collinear if and only if
A
B
=
A
P
AB = AP
A
B
=
A
P
.
5
1
Hide problems
ratio wanted related to orthocenter, circumcenter and 5 midpoints
In triangle
A
B
C
ABC
A
BC
, (
A
B
≠
A
C
AB \ne AC
A
B
=
A
C
), let the orthocenter be
H
H
H
, circumcenter be
O
O
O
, and the midpoint of
B
C
BC
BC
be
M
M
M
. Let
H
M
∩
A
O
=
D
HM \cap AO = D
H
M
∩
A
O
=
D
. Let
P
,
Q
,
R
,
S
P,Q,R,S
P
,
Q
,
R
,
S
be the midpoints of
A
B
,
C
D
,
A
C
,
B
D
AB,CD,AC,BD
A
B
,
C
D
,
A
C
,
B
D
. Let
X
=
P
Q
∩
R
S
X = PQ\cap RS
X
=
PQ
∩
RS
. Find
A
H
/
O
X
AH/OX
A
H
/
OX
.
7
1
Hide problems
A discrete inequality looks like continuous one
For those real numbers
x
1
,
x
2
,
…
,
x
2011
x_1 , x_2 , \ldots , x_{2011}
x
1
,
x
2
,
…
,
x
2011
where each of which satisfies
0
≤
x
1
≤
1
0 \le x_1 \le 1
0
≤
x
1
≤
1
(
i
=
1
,
2
,
…
,
2011
i = 1 , 2 , \ldots , 2011
i
=
1
,
2
,
…
,
2011
), find the maximum of
x
1
3
+
x
2
3
+
⋯
+
x
2011
3
−
(
x
1
x
2
x
3
+
x
2
x
3
x
4
+
⋯
+
x
2011
x
1
x
2
)
x_1^3+x_2^3+ \cdots + x_{2011}^3 - \left( x_1x_2x_3 + x_2x_3x_4 + \cdots + x_{2011}x_1x_2 \right)
x
1
3
+
x
2
3
+
⋯
+
x
2011
3
−
(
x
1
x
2
x
3
+
x
2
x
3
x
4
+
⋯
+
x
2011
x
1
x
2
)
1
1
Hide problems
A three variable equations
Real numbers
a
a
a
,
b
b
b
,
c
c
c
which are differ from
1
1
1
satisfies the following conditions; (1)
a
b
c
=
1
abc =1
ab
c
=
1
(2)
a
2
+
b
2
+
c
2
−
(
1
a
2
+
1
b
2
+
1
c
2
)
=
8
(
a
+
b
+
c
)
−
8
(
a
b
+
b
c
+
c
a
)
a^2+b^2+c^2 - \left( \dfrac{1}{a^2} + \dfrac{1}{b^2} + \dfrac{1}{c^2} \right) = 8(a+b+c) - 8 (ab+bc+ca)
a
2
+
b
2
+
c
2
−
(
a
2
1
+
b
2
1
+
c
2
1
)
=
8
(
a
+
b
+
c
)
−
8
(
ab
+
b
c
+
c
a
)
Find all possible values of expression
1
a
−
1
+
1
b
−
1
+
1
c
−
1
\dfrac{1}{a-1} + \dfrac{1}{b-1} + \dfrac{1}{c-1}
a
−
1
1
+
b
−
1
1
+
c
−
1
1
.