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Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2015 Korea Junior Math Olympiad
2015 Korea Junior Math Olympiad
Part of
Korea Junior Mathematics Olympiad
Subcontests
(7)
8
1
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Pretty much the same as KMO P7
A positive integer
n
n
n
is given. If there exist sets
F
1
,
F
2
,
⋯
F
m
F_1, F_2, \cdots F_m
F
1
,
F
2
,
⋯
F
m
satisfying the following, prove that
m
≤
n
m \le n
m
≤
n
. (For sets
A
,
B
A, B
A
,
B
,
∣
A
∣
|A|
∣
A
∣
is the number of elements in
A
A
A
.
A
−
B
A-B
A
−
B
is the set of elements that are in
A
A
A
but not
B
B
B
)(i): For all
1
≤
i
≤
m
1 \le i \le m
1
≤
i
≤
m
,
F
i
⊆
{
1
,
2
,
⋯
n
}
F_i \subseteq \{1,2,\cdots n\}
F
i
⊆
{
1
,
2
,
⋯
n
}
(ii):
∣
F
1
∣
≤
∣
F
2
∣
≤
⋯
≤
∣
F
m
∣
|F_1| \le |F_2| \le \cdots \le |F_m|
∣
F
1
∣
≤
∣
F
2
∣
≤
⋯
≤
∣
F
m
∣
(iii): For all
1
≤
i
<
j
≤
m
1 \le i < j \le m
1
≤
i
<
j
≤
m
,
∣
F
i
−
F
j
∣
=
1
|F_i-F_j|=1
∣
F
i
−
F
j
∣
=
1
.
7
1
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Prove Schur's Theorem
For a polynomial
f
(
x
)
f(x)
f
(
x
)
with integer coefficients and degree no less than
1
1
1
, prove that there are infinitely many primes
p
p
p
which satisfies the following.There exists an integer
n
n
n
such that
f
(
n
)
≠
0
f(n) \not= 0
f
(
n
)
=
0
and
∣
f
(
n
)
∣
|f(n)|
∣
f
(
n
)
∣
is a multiple of
p
p
p
.
6
1
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Injective Function, FE
Find all functions
f
:
R
→
R
f: \mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
such that (i): For different reals
x
,
y
x,y
x
,
y
,
f
(
x
)
≠
f
(
y
)
f(x) \not= f(y)
f
(
x
)
=
f
(
y
)
.(ii): For all reals
x
,
y
x,y
x
,
y
,
f
(
x
+
f
(
f
(
−
y
)
)
)
=
f
(
x
)
+
f
(
f
(
y
)
)
f(x+f(f(-y)))=f(x)+f(f(y))
f
(
x
+
f
(
f
(
−
y
)))
=
f
(
x
)
+
f
(
f
(
y
))
5
1
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Concurrent Lines - Incenter and Circumcircles
Let
I
I
I
be the incenter of an acute triangle
△
A
B
C
\triangle ABC
△
A
BC
, and let the incircle be
Γ
\Gamma
Γ
. Let the circumcircle of
△
I
B
C
\triangle IBC
△
I
BC
hit
Γ
\Gamma
Γ
at
D
,
E
D, E
D
,
E
, where
D
D
D
is closer to
B
B
B
and
E
E
E
is closer to
C
C
C
. Let
Γ
∩
B
E
=
K
(
≠
E
)
\Gamma \cap BE = K (\not= E)
Γ
∩
BE
=
K
(
=
E
)
,
C
D
∩
B
I
=
T
CD \cap BI = T
C
D
∩
B
I
=
T
, and
C
D
∩
Γ
=
L
(
≠
D
)
CD \cap \Gamma = L (\not= D)
C
D
∩
Γ
=
L
(
=
D
)
. Let the line passing
T
T
T
and perpendicular to
B
I
BI
B
I
meet
Γ
\Gamma
Γ
at
P
P
P
, where
P
P
P
is inside
△
I
B
C
\triangle IBC
△
I
BC
. Prove that the tangent to
Γ
\Gamma
Γ
at
P
P
P
,
K
L
KL
K
L
,
B
I
BI
B
I
are concurrent.
4
1
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Very similar to (actually same) KMO P3
Reals
a
,
b
,
c
,
x
,
y
a,b,c,x,y
a
,
b
,
c
,
x
,
y
satisfy
a
2
+
b
2
+
c
2
=
x
2
+
y
2
=
1
a^2+b^2+c^2=x^2+y^2=1
a
2
+
b
2
+
c
2
=
x
2
+
y
2
=
1
. Find the maximum value of
(
a
x
+
b
y
)
2
+
(
b
x
+
c
y
)
2
(ax+by)^2+(bx+cy)^2
(
a
x
+
b
y
)
2
+
(
b
x
+
cy
)
2
3
1
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Seven Cards for Each Powers of 2
For all nonnegative integer
i
i
i
, there are seven cards with
2
i
2^i
2
i
written on it. How many ways are there to select the cards so that the numbers add up to
n
n
n
?
1
1
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Easy Geometry, Colinear Points
In an acute, scalene triangle
△
A
B
C
\triangle ABC
△
A
BC
, let
O
O
O
be the circumcenter. Let
M
M
M
be the midpoint of
A
C
AC
A
C
. Let the perpendicular from
A
A
A
to
B
C
BC
BC
be
D
D
D
. Let the circumcircle of
△
O
A
M
\triangle OAM
△
O
A
M
hit
D
M
DM
D
M
at
P
(
≠
M
)
P(\not= M)
P
(
=
M
)
. Prove that
B
,
O
,
P
B, O, P
B
,
O
,
P
are colinear.