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Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2006 National Olympiad First Round
9
9
Part of
2006 National Olympiad First Round
Problems
(1)
Turkey NMO 2006 1st Round - P09 (Geometry)
Source:
2/2/2013
A
B
C
ABC
A
BC
is a triangle with
∣
A
B
∣
=
6
|AB|=6
∣
A
B
∣
=
6
,
∣
B
C
∣
=
7
|BC|=7
∣
BC
∣
=
7
, and
∣
A
C
∣
=
8
|AC|=8
∣
A
C
∣
=
8
. Let the angle bisector of
∠
A
\angle A
∠
A
intersect
B
C
BC
BC
at
D
D
D
. If
E
E
E
is a point on
[
A
C
]
[AC]
[
A
C
]
such that
∣
C
E
∣
=
2
|CE|=2
∣
CE
∣
=
2
, what is
∣
D
E
∣
|DE|
∣
D
E
∣
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
17
5
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
7
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
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
3
2
<span class='latex-bold'>(A)</span>\ 3 \qquad<span class='latex-bold'>(B)</span>\ \frac {17}5 \qquad<span class='latex-bold'>(C)</span>\ \frac 72 \qquad<span class='latex-bold'>(D)</span>\ 2\sqrt 3 \qquad<span class='latex-bold'>(E)</span>\ 3\sqrt 2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
5
17
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
2
7
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
2
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
3
2
geometry
angle bisector