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Problems
Contests
National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2003 Korea Junior Math Olympiad
2003 Korea Junior Math Olympiad
Part of
Korea Junior Mathematics Olympiad
Subcontests
(5)
5
1
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2003 KJMO P5 remainder always 1
Four odd positive intgers
a
,
b
,
c
,
d
(
a
≤
b
≤
c
≤
d
)
a, b, c, d (a\leq b \leq c\leq d)
a
,
b
,
c
,
d
(
a
≤
b
≤
c
≤
d
)
are given. Choose any three numbers among them and divide their sum by the un-chosen number, and you will always get the remainder as
1
1
1
. Find all
(
a
,
b
,
c
,
d
)
(a, b, c, d)
(
a
,
b
,
c
,
d
)
that satisfies this.
4
1
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Choosing 6 integers to make 6k
When any
11
11
11
integers are given, prove that you can always choose
6
6
6
integers among them so that the sum of the chosen numbers is a multiple of
6
6
6
. The
11
11
11
integers aren't necessarily different.
3
1
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2003 KJMO P3
Consider a triangle
A
B
C
ABC
A
BC
, inscribed in
O
O
O
and
∠
A
<
∠
B
\angle A < \angle B
∠
A
<
∠
B
. Some point
P
P
P
outside the circle satisfies
∠
A
=
∠
P
B
A
=
18
0
∘
−
∠
P
C
B
\angle A=\angle PBA =180^{\circ}- \angle PCB
∠
A
=
∠
PB
A
=
18
0
∘
−
∠
PCB
Let
D
D
D
be the intersection of line
P
B
PB
PB
and
O
O
O
(different from
B
B
B
), and
Q
Q
Q
the intersection of the tangent line of
O
O
O
passing through
A
A
A
and line
C
D
CD
C
D
. Show that
C
Q
:
A
B
=
A
Q
2
:
A
D
2
CQ : AB=AQ^2:AD^2
CQ
:
A
B
=
A
Q
2
:
A
D
2
.
2
1
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Quadratic equation two integer solution for (a-b)^2<=8sqrt(ab)
a
,
b
a, b
a
,
b
are odd numbers that satisfy
(
a
−
b
)
2
≤
8
a
b
(a-b)^2 \le 8\sqrt {ab}
(
a
−
b
)
2
≤
8
ab
. For
n
=
a
b
n=ab
n
=
ab
, show that the equation
x
2
−
2
(
[
n
]
+
1
)
x
+
n
=
0
x^2-2([\sqrt n]+1)x+n=0
x
2
−
2
([
n
]
+
1
)
x
+
n
=
0
has two integral solutions.
[
r
]
[r]
[
r
]
denotes the biggest integer, not strictly bigger than
r
r
r
.
1
1
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2^2n+1 is not sum of four squares(not 0)
Show that for any non-negative integer
n
n
n
, the number
2
2
n
+
1
2^{2n+1}
2
2
n
+
1
cannot be expressed as a sum of four non-zero square numbers.