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Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2020 Korea Junior Math Olympiad
2020 Korea Junior Math Olympiad
Part of
Korea Junior Mathematics Olympiad
Subcontests
(6)
5
1
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max ae if a+e=1, b+c+d=3, a^2+b^2+c^2+d^2+e^2=14, a<= b<=c<=d<=e
Let
a
,
b
,
c
,
d
,
e
a, b, c, d, e
a
,
b
,
c
,
d
,
e
be real numbers satisfying the following conditions. a \le b \le c \le d \le e, a+e=1, b+c+d=3, a^2+b^2+c^2+d^2+e^2=14Determine the maximum possible value of
a
e
ae
a
e
.
4
1
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collinear wanted, altitudes, <QFD = < EPC, OH _|_ AQ
In an acute triangle
A
B
C
ABC
A
BC
with
A
B
‾
>
A
C
‾
\overline{AB} > \overline{AC}
A
B
>
A
C
, let
D
,
E
,
F
D, E, F
D
,
E
,
F
be the feet of the altitudes from
A
,
B
,
C
A, B, C
A
,
B
,
C
, respectively. Let
P
P
P
be an intersection of lines
E
F
EF
EF
and
B
C
BC
BC
, and let
Q
Q
Q
be a point on the segment
B
D
BD
B
D
such that
∠
Q
F
D
=
∠
E
P
C
\angle QFD = \angle EPC
∠
QF
D
=
∠
EPC
. Let
O
,
H
O, H
O
,
H
denote the circumcenter and the orthocenter of triangle
A
B
C
ABC
A
BC
, respectively. Suppose that
O
H
OH
O
H
is perpendicular to
A
Q
AQ
A
Q
. Prove that
P
,
O
,
H
P, O, H
P
,
O
,
H
are collinear.
2
1
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AK_|_ BC wanted, 2 circles related
Let
A
B
C
ABC
A
BC
be an acute triangle with circumcircle
Ω
\Omega
Ω
and
A
B
‾
<
A
C
‾
\overline{AB} < \overline{AC}
A
B
<
A
C
. The angle bisector of
A
A
A
meets
Ω
\Omega
Ω
again at
D
D
D
, and the line through
D
D
D
, perpendicular to
B
C
BC
BC
meets
Ω
\Omega
Ω
again at
E
E
E
. The circle centered at
A
A
A
, passing through
E
E
E
meets the line
D
E
DE
D
E
again at
F
F
F
. Let
K
K
K
be the circumcircle of triangle
A
D
F
ADF
A
D
F
. Prove that
A
K
AK
A
K
is perpendicular to
B
C
BC
BC
.
6
1
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find the greatest positive integer n
for a positive integer
n
n
n
, there are positive integers
a
1
,
a
2
,
.
.
.
a
n
a_1, a_2, ... a_n
a
1
,
a
2
,
...
a
n
that satisfy these two. (1)
a
1
=
1
,
a
n
=
2020
a_1=1, a_n=2020
a
1
=
1
,
a
n
=
2020
(2) for all integer
i
i
i
,
i
i
i
satisfies
2
≤
i
≤
n
,
a
i
−
a
i
−
1
=
−
2
2\leq i\leq n, a_i-a_{i-1}=-2
2
≤
i
≤
n
,
a
i
−
a
i
−
1
=
−
2
or
3
3
3
. find the greatest
n
n
n
1
1
Hide problems
Number theory problem!!!
The integer n is a number expressed as the sum of an even number of different positive integers less than or equal to 2000. 1+2+ · · · +2000 Find all of the following positive integers that cannot be the value of n.
3
1
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The Number of pairs
The permutation
σ
\sigma
σ
consisting of four words
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
has
f
A
B
(
σ
)
f_{AB}(\sigma)
f
A
B
(
σ
)
, the sum of the number of
B
B
B
placed rightside of every
A
A
A
. We can define
f
B
C
(
σ
)
f_{BC}(\sigma)
f
BC
(
σ
)
,
f
C
D
(
σ
)
f_{CD}(\sigma)
f
C
D
(
σ
)
,
f
D
A
(
σ
)
f_{DA}(\sigma)
f
D
A
(
σ
)
as the same way too. For example,
σ
=
A
C
B
D
B
A
C
D
C
B
A
D
\sigma=ACBDBACDCBAD
σ
=
A
CB
D
B
A
C
D
CB
A
D
,
f
A
B
(
σ
)
=
3
+
1
+
0
=
4
f_{AB}(\sigma)=3+1+0=4
f
A
B
(
σ
)
=
3
+
1
+
0
=
4
,
f
B
C
(
σ
)
=
4
f_{BC}(\sigma)=4
f
BC
(
σ
)
=
4
,
f
C
D
(
σ
)
=
6
f_{CD}(\sigma)=6
f
C
D
(
σ
)
=
6
,
f
D
A
(
σ
)
=
3
f_{DA}(\sigma)=3
f
D
A
(
σ
)
=
3
Find the maximal value of
f
A
B
(
σ
)
+
f
B
C
(
σ
)
+
f
C
D
(
σ
)
+
f
D
A
(
σ
)
f_{AB}(\sigma)+f_{BC}(\sigma)+f_{CD}(\sigma)+f_{DA}(\sigma)
f
A
B
(
σ
)
+
f
BC
(
σ
)
+
f
C
D
(
σ
)
+
f
D
A
(
σ
)
, when
σ
\sigma
σ
consists of
2020
2020
2020
letters for each
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D