MathDB
Problems
Contests
National and Regional Contests
Korea Contests
Korea National Olympiad
2016 Korea National Olympiad
2016 Korea National Olympiad
Part of
Korea National Olympiad
Subcontests
(8)
8
1
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Set differences
A subset
S
∈
{
0
,
1
,
2
,
⋯
,
2000
}
S \in \{0, 1, 2, \cdots , 2000\}
S
∈
{
0
,
1
,
2
,
⋯
,
2000
}
satisfies
∣
S
∣
=
401
|S|=401
∣
S
∣
=
401
. Prove that there exists a positive integer
n
n
n
such that there are at least
70
70
70
positive integers
x
x
x
such that
x
,
x
+
n
∈
S
x, x+n \in S
x
,
x
+
n
∈
S
6
1
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Chebyshev with Max
For a positive integer
n
n
n
, there are
n
n
n
positive reals
a
1
≥
a
2
≥
a
3
⋯
≥
a
n
a_1 \ge a_2 \ge a_3 \cdots \ge a_n
a
1
≥
a
2
≥
a
3
⋯
≥
a
n
. For all positive reals
b
1
,
b
2
,
⋯
b
n
b_1, b_2, \cdots b_n
b
1
,
b
2
,
⋯
b
n
, prove the following inequality.
a
1
b
1
+
a
2
b
2
+
⋯
+
a
n
b
n
a
1
+
a
2
+
⋯
a
n
≤
max
{
b
1
1
,
b
1
+
b
2
2
,
⋯
,
b
1
+
b
2
+
⋯
+
b
n
n
}
\frac{a_1b_1+a_2b_2 + \cdots +a_nb_n}{a_1+a_2+ \cdots a_n} \le \text{max}\{ \frac{b_1}{1}, \frac{b_1+b_2}{2}, \cdots, \frac{b_1+b_2+ \cdots +b_n}{n} \}
a
1
+
a
2
+
⋯
a
n
a
1
b
1
+
a
2
b
2
+
⋯
+
a
n
b
n
≤
max
{
1
b
1
,
2
b
1
+
b
2
,
⋯
,
n
b
1
+
b
2
+
⋯
+
b
n
}
5
1
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Perpendicularity in the Incircle Chord Lemma
A non-isosceles triangle
△
A
B
C
\triangle ABC
△
A
BC
has incenter
I
I
I
and the incircle hits
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
at
D
,
E
,
F
D, E, F
D
,
E
,
F
. Let
E
F
EF
EF
hit the circumcircle of
C
E
I
CEI
CE
I
at
P
≠
E
P \not= E
P
=
E
. Prove that
△
A
B
C
=
2
△
A
B
P
\triangle ABC = 2 \triangle ABP
△
A
BC
=
2△
A
BP
.
4
1
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Number of n-tuples with given condition
For a positive integer
n
n
n
,
S
n
S_n
S
n
is the set of positive integer
n
n
n
-tuples
(
a
1
,
a
2
,
⋯
,
a
n
)
(a_1,a_2, \cdots ,a_n)
(
a
1
,
a
2
,
⋯
,
a
n
)
which satisfies the following.(i).
a
1
=
1
a_1=1
a
1
=
1
.(ii).
a
i
+
1
≤
a
i
+
1
a_{i+1} \le a_i+1
a
i
+
1
≤
a
i
+
1
.For
k
≤
n
k \le n
k
≤
n
, define
N
k
N_k
N
k
as the number of
n
n
n
-tuples
(
a
1
,
a
2
,
⋯
a
n
)
∈
S
n
(a_1, a_2, \cdots a_n) \in S_n
(
a
1
,
a
2
,
⋯
a
n
)
∈
S
n
such that
a
k
=
1
,
a
k
+
1
=
2
a_k=1, a_{k+1}=2
a
k
=
1
,
a
k
+
1
=
2
.Find the sum
N
1
+
N
2
+
⋯
N
k
−
1
N_1 + N_2+ \cdots N_{k-1}
N
1
+
N
2
+
⋯
N
k
−
1
.
3
1
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Weird Ineq
Acute triangle
△
A
B
C
\triangle ABC
△
A
BC
has area
S
S
S
and perimeter
L
L
L
. A point
P
P
P
inside
△
A
B
C
\triangle ABC
△
A
BC
has
d
i
s
t
(
P
,
B
C
)
=
1
,
d
i
s
t
(
P
,
C
A
)
=
1.5
,
d
i
s
t
(
P
,
A
B
)
=
2
dist(P,BC)=1, dist(P,CA)=1.5, dist(P,AB)=2
d
i
s
t
(
P
,
BC
)
=
1
,
d
i
s
t
(
P
,
C
A
)
=
1.5
,
d
i
s
t
(
P
,
A
B
)
=
2
. Let
B
C
∩
A
P
=
D
BC \cap AP = D
BC
∩
A
P
=
D
,
C
A
∩
B
P
=
E
CA \cap BP = E
C
A
∩
BP
=
E
,
A
B
∩
C
P
=
F
AB \cap CP= F
A
B
∩
CP
=
F
. Let
T
T
T
be the area of
△
D
E
F
\triangle DEF
△
D
EF
. Prove the following inequality.
(
A
D
⋅
B
E
⋅
C
F
T
)
2
>
4
L
2
+
(
A
B
⋅
B
C
⋅
C
A
24
S
)
2
\left( \frac{AD \cdot BE \cdot CF}{T} \right)^2 > 4L^2 + \left( \frac{AB \cdot BC \cdot CA}{24S} \right)^2
(
T
A
D
⋅
BE
⋅
CF
)
2
>
4
L
2
+
(
24
S
A
B
⋅
BC
⋅
C
A
)
2
2
1
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EASY GEO
A non-isosceles triangle
△
A
B
C
\triangle ABC
△
A
BC
has its incircle tangent to
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
at points
D
,
E
,
F
D, E, F
D
,
E
,
F
. Let the incenter be
I
I
I
. Say
A
D
AD
A
D
hits the incircle again at
G
G
G
, at let the tangent to the incircle at
G
G
G
hit
A
C
AC
A
C
at
H
H
H
. Let
I
H
∩
A
D
=
K
IH \cap AD = K
I
H
∩
A
D
=
K
, and let the foot of the perpendicular from
I
I
I
to
A
D
AD
A
D
be
L
L
L
.Prove that
I
E
⋅
I
K
=
I
C
⋅
I
L
IE \cdot IK= IC \cdot IL
I
E
⋅
I
K
=
I
C
⋅
I
L
.
7
1
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2016 Korea MO P7 - Simple NT
Let
N
=
2
a
p
1
b
1
p
2
b
2
…
p
k
b
k
N=2^a p_1^{b_1} p_2^{b_2} \ldots p_k^{b_k}
N
=
2
a
p
1
b
1
p
2
b
2
…
p
k
b
k
. Prove that there are
(
b
1
+
1
)
(
b
2
+
1
)
…
(
b
k
+
1
)
(b_1+1)(b_2+1)\ldots(b_k+1)
(
b
1
+
1
)
(
b
2
+
1
)
…
(
b
k
+
1
)
number of
n
n
n
s which satisfies these two conditions.
n
(
n
+
1
)
2
≤
N
\frac{n(n+1)}{2}\le N
2
n
(
n
+
1
)
≤
N
,
N
−
n
(
n
+
1
)
2
N-\frac{n(n+1)}{2}
N
−
2
n
(
n
+
1
)
is divided by
n
n
n
.
1
1
Hide problems
2016 Korea National Olympiad P1-NT
n
n
n
is a positive integer. The number of solutions of
x
2
+
2016
y
2
=
201
7
n
x^2+2016y^2=2017^n
x
2
+
2016
y
2
=
201
7
n
is
k
k
k
. Write
k
k
k
with
n
n
n
.