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Problems
Contests
National and Regional Contests
Korea Contests
Korea National Olympiad
2012 Korea National Olympiad
2012 Korea National Olympiad
Part of
Korea National Olympiad
Subcontests
(4)
1
2
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Korea Second Round 2012 Problem 1
Let
A
B
C
ABC
A
BC
be an obtuse triangle with
∠
A
>
9
0
∘
\angle A > 90^{\circ}
∠
A
>
9
0
∘
. Let circle
O
O
O
be the circumcircle of
A
B
C
ABC
A
BC
.
D
D
D
is a point lying on segment
A
B
AB
A
B
such that
A
D
=
A
C
AD = AC
A
D
=
A
C
. Let
A
K
AK
A
K
be the diameter of circle
O
O
O
. Two lines
A
K
AK
A
K
and
C
D
CD
C
D
meet at
L
L
L
. A circle passing through
D
,
K
,
L
D, K, L
D
,
K
,
L
meets with circle
O
O
O
at
P
(
≠
K
)
P ( \ne K )
P
(
=
K
)
. Given that
A
K
=
2
,
∠
B
C
D
=
∠
B
A
P
=
1
0
∘
AK = 2, \angle BCD = \angle BAP = 10^{\circ}
A
K
=
2
,
∠
BC
D
=
∠
B
A
P
=
1
0
∘
, prove that
D
P
=
sin
(
∠
A
2
)
DP = \sin ( \frac{ \angle A}{2} )
D
P
=
sin
(
2
∠
A
)
.
Consecutive terms of sequence
p
>
3
p >3
p
>
3
is a prime number such that
p
∣
2
p
−
1
−
1
p | 2^{p-1} -1
p
∣
2
p
−
1
−
1
and
p
∤
2
x
−
1
p \not | 2^x - 1
p
∣
2
x
−
1
for
x
=
1
,
2
,
⋯
,
p
−
2
x = 1, 2, \cdots , p-2
x
=
1
,
2
,
⋯
,
p
−
2
. Let
p
=
2
k
+
3
p = 2k+3
p
=
2
k
+
3
. Now we define sequence
{
a
n
}
\{ a_n \}
{
a
n
}
as
a
i
=
a
i
+
k
=
2
i
(
1
≤
i
≤
k
)
,
a
j
+
2
k
=
a
j
a
j
+
k
(
j
≥
1
)
a_i = a_{i+k}= 2^i ( 1 \le i \le k ) , \ a_{j+2k} = a_j a_{j+k} \ ( j \ge 1 )
a
i
=
a
i
+
k
=
2
i
(
1
≤
i
≤
k
)
,
a
j
+
2
k
=
a
j
a
j
+
k
(
j
≥
1
)
Prove that there exist
2
k
2k
2
k
consecutive terms of sequence
a
x
+
1
,
a
x
+
2
,
⋯
,
a
x
+
2
k
a_{x+1} , a_{x+2} , \cdots , a_{x+2k}
a
x
+
1
,
a
x
+
2
,
⋯
,
a
x
+
2
k
such that for all
1
≤
i
<
j
≤
2
k
1 \le i < j \le 2k
1
≤
i
<
j
≤
2
k
,
a
x
+
i
≢
a
x
+
j
(
m
o
d
p
)
a_{x+i} \not \equiv a_{x+j} \ (mod \ p)
a
x
+
i
≡
a
x
+
j
(
m
o
d
p
)
.
4
2
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Maximum with condition a^2 + b^2 + c^2 = 2abc + 1
a
,
b
,
c
a,b,c
a
,
b
,
c
are positive numbers such that
a
2
+
b
2
+
c
2
=
2
a
b
c
+
1
a^2 + b^2 + c^2 = 2abc + 1
a
2
+
b
2
+
c
2
=
2
ab
c
+
1
. Find the maximum value of
(
a
−
2
b
c
)
(
b
−
2
c
a
)
(
c
−
2
a
b
)
(a-2bc)(b-2ca)(c-2ab)
(
a
−
2
b
c
)
(
b
−
2
c
a
)
(
c
−
2
ab
)
Large subset of pairs satisfying a divisibility condition
Let
p
≡
3
(
m
o
d
4
)
p \equiv 3 \pmod{4}
p
≡
3
(
mod
4
)
be a prime. Define
T
=
{
(
i
,
j
)
∣
i
,
j
∈
{
0
,
1
,
⋯
,
p
−
1
}
}
∖
{
(
0
,
0
)
}
T = \{ (i,j) \mid i, j \in \{ 0, 1, \cdots , p-1 \} \} \smallsetminus \{ (0,0) \}
T
=
{(
i
,
j
)
∣
i
,
j
∈
{
0
,
1
,
⋯
,
p
−
1
}}
∖
{(
0
,
0
)}
. For arbitrary subset
S
(
≠
∅
)
⊂
T
S ( \ne \emptyset ) \subset T
S
(
=
∅
)
⊂
T
, prove that there exist subset
A
⊂
S
A \subset S
A
⊂
S
satisfying following conditions: (a)
(
x
i
,
y
i
)
∈
A
(
1
≤
i
≤
3
)
(x_i , y_i ) \in A ( 1 \le i \le 3)
(
x
i
,
y
i
)
∈
A
(
1
≤
i
≤
3
)
then
p
∤
x
1
+
x
2
−
y
3
p \not | x_1 + x_2 - y_3
p
∣
x
1
+
x
2
−
y
3
or
p
∤
y
1
+
y
2
+
x
3
p \not | y_1 + y_2 + x_3
p
∣
y
1
+
y
2
+
x
3
. (b)
8
n
(
A
)
>
n
(
S
)
8 n(A) > n(S)
8
n
(
A
)
>
n
(
S
)
3
2
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Diophantine Equation with prime numbers
Find all triples
(
m
,
p
,
q
)
(m,p,q)
(
m
,
p
,
q
)
where
m
m
m
is a positive integer and
p
,
q
p , q
p
,
q
are primes.
2
m
p
2
+
1
=
q
5
2^m p^2 + 1 = q^5
2
m
p
2
+
1
=
q
5
Maximum value in permutations
Let
{
a
1
,
a
2
,
⋯
,
a
10
}
=
{
1
,
2
,
⋯
,
10
}
\{ a_1 , a_2 , \cdots, a_{10} \} = \{ 1, 2, \cdots , 10 \}
{
a
1
,
a
2
,
⋯
,
a
10
}
=
{
1
,
2
,
⋯
,
10
}
. Find the maximum value of
∑
n
=
1
10
(
n
a
n
2
−
n
2
a
n
)
\sum_{n=1}^{10}(na_n ^2 - n^2 a_n )
n
=
1
∑
10
(
n
a
n
2
−
n
2
a
n
)
2
2
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An inequality at a graph
There are
n
n
n
students
A
1
,
A
2
,
⋯
,
A
n
A_1 , A_2 , \cdots , A_n
A
1
,
A
2
,
⋯
,
A
n
and some of them shaked hands with each other. (
A
i
A_i
A
i
and
A
j
A_j
A
j
can shake hands more than one time.) Let the student
A
i
A_i
A
i
shaked hands
d
i
d_i
d
i
times. Suppose
d
1
+
d
2
+
⋯
+
d
n
>
0
d_1 + d_2 + \cdots + d_n > 0
d
1
+
d
2
+
⋯
+
d
n
>
0
. Prove that there exist
1
≤
i
<
j
≤
n
1 \le i < j \le n
1
≤
i
<
j
≤
n
satisfying the following conditions: (a) Two students
A
i
A_i
A
i
and
A
j
A_j
A
j
shaked hands each other. (b)
(
d
1
+
d
2
+
⋯
+
d
n
)
2
n
2
≤
d
i
d
j
\frac{(d_1 + d_2 + \cdots + d_n ) ^2 }{n^2 } \le d_i d_j
n
2
(
d
1
+
d
2
+
⋯
+
d
n
)
2
≤
d
i
d
j
Parallel lines
Let
w
w
w
be the incircle of triangle
A
B
C
ABC
A
BC
. Segments
B
C
,
C
A
BC, CA
BC
,
C
A
meet with
w
w
w
at points
D
,
E
D, E
D
,
E
. A line passing through
B
B
B
and parallel to
D
E
DE
D
E
meets
w
w
w
at
F
F
F
and
G
G
G
. (
F
F
F
is nearer to
B
B
B
than
G
G
G
.) Line
C
G
CG
CG
meets
w
w
w
at
H
(
≠
G
)
H ( \ne G )
H
(
=
G
)
. A line passing through
G
G
G
and parallel to
E
H
EH
E
H
meets with line
A
C
AC
A
C
at
I
I
I
. Line
I
F
IF
I
F
meets with circle
w
w
w
at
J
(
≠
F
)
J (\ne F )
J
(
=
F
)
. Lines
C
J
CJ
C
J
and
E
G
EG
EG
meets at
K
K
K
. Let
l
l
l
be the line passing through
K
K
K
and parallel to
J
D
JD
J
D
. Prove that
l
,
I
F
,
E
D
l, IF, ED
l
,
I
F
,
E
D
meet at one point.