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Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2014 National Olympiad First Round
5
5
Part of
2014 National Olympiad First Round
Problems
(1)
P05 [Geometry] - Turkish NMO 1st Round - 2014
Source:
5/21/2014
Let
D
D
D
be a point on side
[
B
C
]
[BC]
[
BC
]
of
△
A
B
C
\triangle ABC
△
A
BC
such that
∣
A
B
∣
=
∣
A
C
∣
|AB|=|AC|
∣
A
B
∣
=
∣
A
C
∣
,
∣
B
D
∣
=
6
|BD|=6
∣
B
D
∣
=
6
and
∣
D
C
∣
=
10
|DC|=10
∣
D
C
∣
=
10
. If the incircles of
△
A
B
D
\triangle ABD
△
A
B
D
and
△
A
D
C
\triangle ADC
△
A
D
C
touch side
[
A
D
]
[AD]
[
A
D
]
at
E
E
E
and
F
F
F
, respectively, what is
∣
E
F
∣
|EF|
∣
EF
∣
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
1
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
2
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
9
8
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
2
<span class='latex-bold'>(A)</span>\ \dfrac{1}{\sqrt{2}} \qquad<span class='latex-bold'>(B)</span>\ \dfrac{2}{\sqrt{3}} \qquad<span class='latex-bold'>(C)</span>\ 1 \qquad<span class='latex-bold'>(D)</span>\ \dfrac{9}{8} \qquad<span class='latex-bold'>(E)</span>\ 2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
2
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
3
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
8
9
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
2
geometry