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National and Regional Contests
Greece Contests
Greece Junior Math Olympiad
1998 Greece Junior Math Olympiad
1998 Greece Junior Math Olympiad
Part of
Greece Junior Math Olympiad
Subcontests
(4)
4
1
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Greece Junior National Olympiad 1997-98 Pronlem 4
Let
K
(
O
,
R
)
K(O,R)
K
(
O
,
R
)
be a circle with center
O
O
O
and radious
R
R
R
and
(
e
)
(e)
(
e
)
to be a line thst tangent to
K
K
K
at
A
A
A
. A line parallel to
O
A
OA
O
A
cuts
K
K
K
at
B
,
C
B, C
B
,
C
, and
(
e
)
(e)
(
e
)
at
D
D
D
, (
C
C
C
is between
B
B
B
and
D
D
D
). Let
E
E
E
to be the antidiameric of
C
C
C
with respect to
K
K
K
.
E
A
EA
E
A
cuts
B
D
BD
B
D
at
F
F
F
.i)Examine if
C
E
F
CEF
CEF
is isosceles. ii)Prove that
2
A
D
=
E
B
2AD=EB
2
A
D
=
EB
. iii)If
K
K
K
si the midlpoint of
C
F
CF
CF
, prove that
A
B
=
K
O
AB=KO
A
B
=
K
O
. iv)If
R
=
5
2
,
A
D
=
3
2
R=\frac{5}{2}, AD=\frac{3}{2}
R
=
2
5
,
A
D
=
2
3
, calculate the area of
E
B
F
EBF
EBF
3
1
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Greece Junior National Olympiad 1977-78 problem 3
Let
k
k
k
be a prime, such as
k
≠
2
,
5
k\neq 2, 5
k
=
2
,
5
, prove that between the first
k
k
k
terms of the sequens
1
,
11
,
111
,
1111
,
.
.
.
.
,
1111....1
1, 11, 111, 1111,....,1111....1
1
,
11
,
111
,
1111
,
....
,
1111....1
, where the last term have
k
k
k
ones, is divisible by
k
k
k
.
2
1
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Greece Junior National Olympiad 1997-98 problem 2
If
a
1
,
a
2
,
.
.
.
.
,
a
n
−
1
,
a
n
a_1, a_2,...., a_{n-1}, a_n
a
1
,
a
2
,
....
,
a
n
−
1
,
a
n
, are positive integers, prove that:
∏
i
=
1
n
(
a
i
2
+
3
a
i
+
1
)
a
1
a
2
.
.
.
.
a
n
−
1
a
n
≥
2
2
n
\frac{\prod_{i=1}^n(a_i^2+3a_i+1)}{a_1a_2....a_{n-1}a_n}\ge 2^{2n}
a
1
a
2
....
a
n
−
1
a
n
∏
i
=
1
n
(
a
i
2
+
3
a
i
+
1
)
≥
2
2
n
1
1
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Greece Junior National Olympiad 1997-1998
Find all he positive integers
x
,
y
,
z
,
t
,
w
x, y, z, t, w
x
,
y
,
z
,
t
,
w
, such as:
x
+
1
y
+
1
z
+
1
t
+
1
w
=
1998
115
x+\frac{1}{y+\frac{1}{z+\frac{1}{t+\frac{1}{w}}}}=\frac{1998}{115}
x
+
y
+
z
+
t
+
w
1
1
1
1
=
115
1998