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Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2000 Korea Junior Math Olympiad
2000 Korea Junior Math Olympiad
Part of
Korea Junior Mathematics Olympiad
Subcontests
(8)
8
1
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Derangement Problem KJMO 2000 P8
n
n
n
men and one woman are in the meeting room with
n
+
1
n+1
n
+
1
chairs, each of them having their own seat. Show that the following two number of cases are equal.(1) Number of cases to choose one man to get out of the room, and make the left
n
−
1
n-1
n
−
1
men to sit to each other's chair. (2) Number of cases to make
n
+
1
n+1
n
+
1
people to sit to each other's chair.
7
1
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2000 KJMO P7 Bashing Geometry
A
B
C
ABC
A
BC
is a triangle that
2
∠
B
<
∠
A
<
9
0
∘
2\angle B < \angle A <90^{\circ}
2∠
B
<
∠
A
<
9
0
∘
, and
P
P
P
is a point on
A
B
AB
A
B
satisfying
∠
A
=
2
∠
A
P
C
\angle A=2\angle APC
∠
A
=
2∠
A
PC
. If
B
C
=
a
BC=a
BC
=
a
,
A
C
=
b
AC=b
A
C
=
b
,
B
P
=
1
BP=1
BP
=
1
, express
A
P
AP
A
P
as a function of
a
,
b
a, b
a
,
b
.
6
1
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2000 KJMO if product is not smaller than the sum
x
,
y
,
z
x, y, z
x
,
y
,
z
are positive reals which their product is not smaller then their sum. Prove the inequality:
2
x
2
+
y
z
+
2
y
2
+
z
x
+
2
z
2
+
x
y
≥
9
\sqrt{2x^2+yz}+\sqrt{2y^2+zx}+\sqrt{2z^2+xy} \geq 9
2
x
2
+
yz
+
2
y
2
+
z
x
+
2
z
2
+
x
y
≥
9
5
1
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a=2(A)99&hellip;99(B)(C)
a
a
a
is a
2000
2000
2000
digit natural number of the form
a
=
2
(
A
)
99
…
99
(
B
)
(
C
)
a=2(A)99…99(B)(C)
a
=
2
(
A
)
99
…
99
(
B
)
(
C
)
expressed in base
10
10
10
.
a
a
a
is not a multiple of
10
10
10
, and
2
(
A
)
+
(
B
)
(
C
)
=
99
2(A)+(B)(C)=99
2
(
A
)
+
(
B
)
(
C
)
=
99
.
a
=
2899..9971
a=2899..9971
a
=
2899..9971
is a possible example of
a
a
a
.
b
b
b
is a number you earn when you write the digits of
a
a
a
in a reverse order(Writing the digits of some number in a reverse order means like reordering
1234
1234
1234
into
4321
4321
4321
). Find every positive integer
a
a
a
that makes
a
b
ab
ab
a square number.
4
1
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KJMO 2000 inequality
Show that for real variables
1
≤
a
,
b
≤
2
1 \leq a, b \leq 2
1
≤
a
,
b
≤
2
the following inequality holds.
2
(
a
+
b
)
2
≤
9
a
b
2(a+b)^2 \leq 9ab
2
(
a
+
b
)
2
≤
9
ab
3
1
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KJMO 2000 P3 AD, BQ, CR meet at one point
Acute triangle
A
B
C
ABC
A
BC
is inscribed in circle
O
O
O
.
P
P
P
is the foot of altitude from
A
A
A
to
B
C
BC
BC
, and
D
D
D
is the intersection of
O
O
O
and line
A
P
AP
A
P
.
M
,
N
M, N
M
,
N
are midpoint of
A
B
,
A
C
AB, AC
A
B
,
A
C
respectively.
M
P
MP
MP
and
C
D
CD
C
D
intersects at
Q
Q
Q
, and
N
P
NP
NP
and
B
D
BD
B
D
intersects at
R
R
R
. Show that
A
D
,
B
Q
,
C
R
AD, BQ, CR
A
D
,
BQ
,
CR
meet at one point if and only if
A
B
=
A
C
AB=AC
A
B
=
A
C
.
2
1
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KJMO 2000 P2 Counting Calendars (again!)
Along consecutive seven days, from Sunday to Saturday, let us call the days belonging to the same month a MB. For example, if the last day of a month is Sunday, the last MB of that month consists of the last day of that month. If a year is from January first to December
31
31
31
, find the maximum and minimum values of MB in one year.
1
1
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2000 KJMO P1 easy euclidean lemma
For arbitrary natural number
a
a
a
, show that
gcd
(
a
3
+
1
,
a
7
+
1
)
=
a
+
1
\gcd(a^3+1, a^7+1)=a+1
g
cd
(
a
3
+
1
,
a
7
+
1
)
=
a
+
1
.