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Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2001 Korea Junior Math Olympiad
2001 Korea Junior Math Olympiad
Part of
Korea Junior Mathematics Olympiad
Subcontests
(8)
8
1
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2001 KJMO P8 distance between a line and a point
A
B
C
D
ABCD
A
BC
D
is a convex quadrilateral, both
∠
A
B
C
\angle ABC
∠
A
BC
and
∠
B
C
D
\angle BCD
∠
BC
D
acute.
E
E
E
is a point inside
A
B
C
D
ABCD
A
BC
D
satisfying
A
E
=
D
E
AE=DE
A
E
=
D
E
, and
X
,
Y
X, Y
X
,
Y
are the intersection of
A
D
AD
A
D
and
C
E
,
B
E
CE, BE
CE
,
BE
respectively, but not
X
=
A
X=A
X
=
A
or
Y
=
D
Y=D
Y
=
D
. If
A
B
E
X
ABEX
A
BEX
and
C
D
E
Y
CDEY
C
D
E
Y
are both inscribed quadrilaterals, prove that the distance between
E
E
E
and the lines
A
B
,
B
C
,
C
D
AB, BC, CD
A
B
,
BC
,
C
D
are all equal.
7
1
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2001 KJMO P7 (arrangement difference sum = n^2)
Finite set
{
a
1
,
a
2
,
.
.
.
,
a
n
,
b
1
,
b
2
,
.
.
.
,
b
n
}
=
{
1
,
2
,
…
,
2
n
}
\{a_1, a_2, ..., a_n, b_1, b_2, ..., b_n\}=\{1, 2, …, 2n\}
{
a
1
,
a
2
,
...
,
a
n
,
b
1
,
b
2
,
...
,
b
n
}
=
{
1
,
2
,
…
,
2
n
}
is given. If
a
1
<
a
2
<
.
.
.
<
a
n
a_1<a_2<...<a_n
a
1
<
a
2
<
...
<
a
n
and
b
1
>
b
2
>
.
.
.
>
b
n
b_1>b_2>...>b_n
b
1
>
b
2
>
...
>
b
n
, show that
∑
i
=
1
n
∣
a
i
−
b
i
∣
=
n
2
\sum_{i=1}^n |a_i-b_i|=n^2
i
=
1
∑
n
∣
a
i
−
b
i
∣
=
n
2
6
1
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2001 KJMO P6
For real variables
0
≤
x
,
y
,
z
,
w
≤
1
0 \leq x, y, z, w \leq 1
0
≤
x
,
y
,
z
,
w
≤
1
, find the maximum value of
x
(
1
−
y
)
+
2
y
(
1
−
z
)
+
3
z
(
1
−
w
)
+
4
w
(
1
−
x
)
x(1-y)+2y(1-z)+3z(1-w)+4w(1-x)
x
(
1
−
y
)
+
2
y
(
1
−
z
)
+
3
z
(
1
−
w
)
+
4
w
(
1
−
x
)
5
1
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2001 KJMO P5
A
A
A
is a set satisfying the following the condition. Show that
2001
+
2001
2001+\sqrt{2001}
2001
+
2001
is an element of
A
A
A
.Condition (1)
1
∈
A
1 \in A
1
∈
A
(2) If
x
∈
A
x \in A
x
∈
A
, then
x
2
∈
A
x^2 \in A
x
2
∈
A
.(3) If
(
x
−
3
)
2
∈
A
(x-3)^2 \in A
(
x
−
3
)
2
∈
A
, then
x
∈
A
x \in A
x
∈
A
.
4
1
Hide problems
2001 KJMO P4
Some
n
≥
3
n \geq 3
n
≥
3
cities are connected with railways, so that you can travel from one city to every other, not necessarily directly. However, the railways are structured in such a way that there is only one way to get from one city to another, assuming you don't pass through the same city again. Let
A
A
A
be the set of these cities and railways. Show that there exists a Subset of
A
A
A
, let's say
C
C
C
, such that(1)
C
C
C
has at least
[
(
n
+
1
)
/
2
]
[(n+1)/2]
[(
n
+
1
)
/2
]
cities as its element.(2) No two elements of
C
C
C
are directly connected with railways.
3
1
Hide problems
2001 KJMO P3
A
,
B
A, B
A
,
B
are points on circle
O
O
O
satisfying
∠
A
O
B
<
12
0
∘
\angle AOB < 120^{\circ}
∠
A
OB
<
12
0
∘
.
C
C
C
is a point on the tangent line of
O
O
O
passing through
A
A
A
satisfying
A
B
=
A
C
AB=AC
A
B
=
A
C
and
∠
B
A
C
<
9
0
∘
\angle BAC < 90^{\circ}
∠
B
A
C
<
9
0
∘
.
D
D
D
is the intersection of
O
O
O
and
B
C
BC
BC
not
B
B
B
, and
I
I
I
is the incenter of
A
B
D
ABD
A
B
D
. Prove that
A
E
=
A
C
AE=AC
A
E
=
A
C
where
E
E
E
is the intersection of
C
I
CI
C
I
and
A
D
AD
A
D
.
2
1
Hide problems
KJMO 2001 P2
n
n
n
is a product of some two consecutive primes.
s
(
n
)
s(n)
s
(
n
)
denotes the sum of the divisors of
n
n
n
and
p
(
n
)
p(n)
p
(
n
)
denotes the number of relatively prime positive integers not exceeding
n
n
n
. Express
s
(
n
)
p
(
n
)
s(n)p(n)
s
(
n
)
p
(
n
)
as a polynomial of
n
n
n
.
1
1
Hide problems
KJMO 2001 P1
A right triangle of the following condition is given: the three side lengths are all positive integers and the length of the shortest segment is
141
141
141
. For the triangle that has the minimum area while satisfying the condition, find the lengths of the other two sides.