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Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2017 Korea Junior Math Olympiad
2017 Korea Junior Math Olympiad
Part of
Korea Junior Mathematics Olympiad
Subcontests
(7)
8
1
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Maximum number of complete graph
For a positive integer
n
n
n
, there is a school with
n
n
n
people. For a set
X
X
X
of students in this school, if any two students in
X
X
X
know each other, we call
X
X
X
well-formed. If the maximum number of students in a well-formed set is
k
k
k
, show that the maximum number of well-formed sets is not greater than
3
(
n
+
k
)
/
3
3^{(n+k)/3}
3
(
n
+
k
)
/3
.Here, an empty set and a set with one student is regarded as well-formed as well.
7
1
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No function satisfying the condition
Prove that there is no function
f
:
R
≥
0
→
R
f:\mathbb{R}_{\ge0}\rightarrow\mathbb{R}
f
:
R
≥
0
→
R
satisfying:
f
(
x
+
y
2
)
≥
f
(
x
)
+
y
f(x+y^2)\ge f(x)+y
f
(
x
+
y
2
)
≥
f
(
x
)
+
y
for all two nonnegative real numbers
x
,
y
x,y
x
,
y
.
6
1
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Circumcircle of medial triangle
Let triangle
A
B
C
ABC
A
BC
be an acute scalene triangle, and denote
D
,
E
,
F
D,E,F
D
,
E
,
F
by the midpoints of
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
, respectively. Let the circumcircle of
D
E
F
DEF
D
EF
be
O
1
O_1
O
1
, and its center be
N
N
N
. Let the circumcircle of
B
C
N
BCN
BCN
be
O
2
O_2
O
2
.
O
1
O_1
O
1
and
O
2
O_2
O
2
meet at two points
P
,
Q
P, Q
P
,
Q
.
O
2
O_2
O
2
meets
A
B
AB
A
B
at point
K
(
≠
B
)
K(\neq B)
K
(
=
B
)
and meets
A
C
AC
A
C
at point
L
(
≠
C
)
L(\neq C)
L
(
=
C
)
. Show that the three lines
E
F
,
P
Q
,
K
L
EF,PQ,KL
EF
,
PQ
,
K
L
are concurrent.
5
1
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m^3+am+b is not a multiple of n
Given an integer
n
≥
2
n\ge 2
n
≥
2
, show that there exist two integers
a
,
b
a,b
a
,
b
which satisfy the following.For all integer
m
m
m
,
m
3
+
a
m
+
b
m^3+am+b
m
3
+
am
+
b
is not a multiple of
n
n
n
.
3
1
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Divisors of 15^25+1
Find all
n
>
1
n>1
n
>
1
and integers
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\dots,a_n
a
1
,
a
2
,
…
,
a
n
satisfying the following three conditions: (i)
2
<
a
1
≤
a
2
≤
⋯
≤
a
n
2<a_1\le a_2\le \cdots\le a_n
2
<
a
1
≤
a
2
≤
⋯
≤
a
n
(ii)
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\dots,a_n
a
1
,
a
2
,
…
,
a
n
are divisors of
1
5
25
+
1
15^{25}+1
1
5
25
+
1
. (iii)
2
−
2
1
5
25
+
1
=
(
1
−
2
a
1
)
+
(
1
−
2
a
2
)
+
⋯
+
(
1
−
2
a
n
)
2-\frac{2}{15^{25}+1}=\left(1-\frac{2}{a_1}\right)+\left(1-\frac{2}{a_2}\right)+\cdots+\left(1-\frac{2}{a_n}\right)
2
−
1
5
25
+
1
2
=
(
1
−
a
1
2
)
+
(
1
−
a
2
2
)
+
⋯
+
(
1
−
a
n
2
)
1
1
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i divides exactly a_i numbers
Find all positive integer
n
n
n
and nonnegative integer
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\dots,a_n
a
1
,
a
2
,
…
,
a
n
satisfying:
i
i
i
divides exactly
a
i
a_i
a
i
numbers among
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\dots,a_n
a
1
,
a
2
,
…
,
a
n
, for each
i
=
1
,
2
,
…
,
n
i=1,2,\dots,n
i
=
1
,
2
,
…
,
n
. (
0
0
0
is divisible by all integers.)
4
1
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An algebraic inequality for 4 variables
4. Let
a
≥
b
≥
c
≥
d
>
0
a \geq b \geq c \geq d>0
a
≥
b
≥
c
≥
d
>
0
. Show that
b
3
a
+
c
3
b
+
d
3
c
+
a
3
d
+
3
(
a
b
+
b
c
+
c
d
+
d
a
)
≥
4
(
a
2
+
b
2
+
c
2
+
d
2
)
.
\frac{b^3}{a} + \frac{c^3}{b} + \frac{d^3}{c} + \frac{a^3}{d} + 3 \left( ab+bc+cd+da \right) \geq 4 {\left( a^2 + b^2 + c^2 +d^2 \right)}.
a
b
3
+
b
c
3
+
c
d
3
+
d
a
3
+
3
(
ab
+
b
c
+
c
d
+
d
a
)
≥
4
(
a
2
+
b
2
+
c
2
+
d
2
)
.
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