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National and Regional Contests
Turkey Contests
National Olympiad First Round
2003 National Olympiad First Round
1
1
Part of
2003 National Olympiad First Round
Problems
(1)
P01 [Geometry] - Turkish NMO 1st Round - 2003
Source:
5/6/2014
Let
A
B
C
ABC
A
BC
be a triangle such that
∣
A
B
∣
=
7
|AB|=7
∣
A
B
∣
=
7
,
∣
B
C
∣
=
8
|BC|=8
∣
BC
∣
=
8
,
∣
A
C
∣
=
6
|AC|=6
∣
A
C
∣
=
6
. Let
D
D
D
be the midpoint of side
[
B
C
]
[BC]
[
BC
]
. If the circle through
A
A
A
,
B
B
B
and
D
D
D
cuts
A
C
AC
A
C
at
A
A
A
and
E
E
E
, what is
∣
A
E
∣
|AE|
∣
A
E
∣
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
2
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
3
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
3
<span class='latex-bold'>(A)</span>\ \dfrac 23 \qquad<span class='latex-bold'>(B)</span>\ 1 \qquad<span class='latex-bold'>(C)</span>\ \dfrac 32 \qquad<span class='latex-bold'>(D)</span>\ 2 \qquad<span class='latex-bold'>(E)</span>\ 3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
3
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
2
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
3
geometry
power of a point