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Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2008 National Olympiad First Round
13
13
Part of
2008 National Olympiad First Round
Problems
(1)
Turkey NMO 2008 1st Round - P13 (Geometry)
Source:
8/25/2012
Let
A
B
C
ABC
A
BC
be a triangle such that angle
C
C
C
is obtuse. Let
D
∈
[
A
B
]
D\in [AB]
D
∈
[
A
B
]
and
[
D
C
]
⊥
[
B
C
]
[DC]\perp [BC]
[
D
C
]
⊥
[
BC
]
. If
m
(
A
B
C
^
)
=
α
m(\widehat{ABC})=\alpha
m
(
A
BC
)
=
α
,
m
(
B
C
A
^
)
=
3
α
m(\widehat{BCA})=3\alpha
m
(
BC
A
)
=
3
α
, and
∣
A
C
∣
−
∣
A
D
∣
=
10
|AC|-|AD|=10
∣
A
C
∣
−
∣
A
D
∣
=
10
, what is
∣
B
D
∣
|BD|
∣
B
D
∣
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
10
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
14
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
18
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
20
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
22
<span class='latex-bold'>(A)</span>\ 10 \qquad<span class='latex-bold'>(B)</span>\ 14 \qquad<span class='latex-bold'>(C)</span>\ 18 \qquad<span class='latex-bold'>(D)</span>\ 20 \qquad<span class='latex-bold'>(E)</span>\ 22
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
10
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
14
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
18
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
20
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
22
geometry
trigonometry