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Greece Junior Math Olympiad
1998 Greece Junior Math Olympiad
4
4
Part of
1998 Greece Junior Math Olympiad
Problems
(1)
Greece Junior National Olympiad 1997-98 Pronlem 4
Source:
8/10/2015
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
geometry