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Problems
Contests
National and Regional Contests
Korea Contests
Korea National Olympiad
2011 Korea National Olympiad
2011 Korea National Olympiad
Part of
Korea National Olympiad
Subcontests
(4)
4
2
Hide problems
Korea Second Round 2011
Let
k
,
n
k,n
k
,
n
be positive integers. There are
k
n
kn
kn
points
P
1
,
P
2
,
⋯
,
P
k
n
P_1, P_2, \cdots, P_{kn}
P
1
,
P
2
,
⋯
,
P
kn
on a circle. We can color each points with one of color
c
1
,
c
2
,
⋯
,
c
k
c_1, c_2, \cdots , c_k
c
1
,
c
2
,
⋯
,
c
k
. In how many ways we can color the points satisfying the following conditions?(a) Each color is used
n
n
n
times.(b)
∀
i
≠
j
\forall i \not = j
∀
i
=
j
, if
P
a
P_a
P
a
and
P
b
P_b
P
b
is colored with color
c
i
c_i
c
i
, and
P
c
P_c
P
c
and
P
d
P_d
P
d
is colored with color
c
j
c_j
c
j
, then the segment
P
a
P
b
P_a P_b
P
a
P
b
and segment
P
c
P
d
P_c P_d
P
c
P
d
doesn't meet together.
Korea Second Round 2011
Let
x
1
,
x
2
,
⋯
,
x
25
x_1, x_2, \cdots, x_{25}
x
1
,
x
2
,
⋯
,
x
25
real numbers such that
0
≤
x
i
≤
i
(
i
=
1
,
2
,
⋯
,
25
)
0 \le x_i \le i (i=1, 2, \cdots, 25)
0
≤
x
i
≤
i
(
i
=
1
,
2
,
⋯
,
25
)
. Find the maximum value of
x
1
3
+
x
2
3
+
⋯
+
x
25
3
−
(
x
1
x
2
x
3
+
x
2
x
3
x
4
+
⋯
x
25
x
1
x
2
)
x_{1}^{3}+x_{2}^{3}+\cdots +x_{25}^{3} - ( x_1x_2x_3 + x_2x_3x_4 + \cdots x_{25}x_1x_2 )
x
1
3
+
x
2
3
+
⋯
+
x
25
3
−
(
x
1
x
2
x
3
+
x
2
x
3
x
4
+
⋯
x
25
x
1
x
2
)
3
2
Hide problems
Korea Second Round 2011
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
real numbers such that
a
+
b
+
c
+
d
=
19
a+b+c+d=19
a
+
b
+
c
+
d
=
19
and
a
2
+
b
2
+
c
2
+
d
2
=
91
a^2+b^2+c^2+d^2=91
a
2
+
b
2
+
c
2
+
d
2
=
91
. Find the maximum value of
1
a
+
1
b
+
1
c
+
1
d
\frac{1}{a} +\frac{1}{b} +\frac{1}{c} +\frac{1}{d}
a
1
+
b
1
+
c
1
+
d
1
Korea Second Round 2011
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
l
l
such that
(
m
−
1
)
!
<
l
<
(
m
+
1
)
!
+
1
(m-1)! < l < (m+1)!+1
(
m
−
1
)!
<
l
<
(
m
+
1
)!
+
1
2
2
Hide problems
Korea Second Round 2011
Let
x
,
y
x, y
x
,
y
be positive integers such that
gcd
(
x
,
y
)
=
1
\gcd(x,y)=1
g
cd
(
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.
Korea Second Round 2011
Let
A
B
C
ABC
A
BC
be a triangle and its incircle meets
B
C
,
A
C
,
A
B
BC, AC, AB
BC
,
A
C
,
A
B
at
D
,
E
D, E
D
,
E
and
F
F
F
respectively. Let point
P
P
P
on the incircle and inside
△
A
E
F
\triangle AEF
△
A
EF
. Let
X
=
P
B
∩
D
F
,
Y
=
P
C
∩
D
E
,
Q
=
E
X
∩
F
Y
X=PB \cap DF , Y=PC \cap DE, Q=EX \cap FY
X
=
PB
∩
D
F
,
Y
=
PC
∩
D
E
,
Q
=
EX
∩
F
Y
. Prove that the points
A
A
A
and
Q
Q
Q
lies on
D
P
DP
D
P
simultaneously or located opposite sides from
D
P
DP
D
P
.
1
2
Hide problems
Korea Second Round 2011
Two circles
O
,
O
′
O, O'
O
,
O
′
having same radius meet at two points,
A
,
B
(
A
≠
B
)
A,B (A \not = B)
A
,
B
(
A
=
B
)
. Point
P
,
Q
P,Q
P
,
Q
are each on circle
O
O
O
and
O
′
O'
O
′
(
P
≠
A
,
B
Q
≠
A
,
B
)
(P \not = A,B ~ Q\not = A,B )
(
P
=
A
,
B
Q
=
A
,
B
)
. Select the point
R
R
R
such that
P
A
Q
R
PAQR
P
A
QR
is a parallelogram. Assume that
B
,
R
,
P
,
Q
B, R, P, Q
B
,
R
,
P
,
Q
is cyclic. Now prove that
P
Q
=
O
O
′
PQ = OO'
PQ
=
O
O
′
.
Korea Second Round 2011
Find the number of positive integer
n
<
3
8
n < 3^8
n
<
3
8
satisfying the following condition."The number of positive integer
k
(
1
≤
k
≤
n
3
)
k (1 \leq k \leq \frac {n}{3})
k
(
1
≤
k
≤
3
n
)
such that
n
!
(
n
−
3
k
)
!
⋅
k
!
⋅
3
k
+
1
\frac{n!}{(n-3k)! \cdot k! \cdot 3^{k+1}}
(
n
−
3
k
)!
⋅
k
!
⋅
3
k
+
1
n
!
is not a integer" is
216
216
216
.