MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2010 National Olympiad First Round
5
5
Part of
2010 National Olympiad First Round
Problems
(1)
Turkey NMO 2010 1st Round - P05 (Geometry)
Source:
7/26/2012
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral such that
∣
A
B
∣
=
10
|AB|=10
∣
A
B
∣
=
10
,
∣
C
D
∣
=
3
6
|CD|=3\sqrt 6
∣
C
D
∣
=
3
6
,
m
(
A
B
D
^
)
=
6
0
∘
m(\widehat{ABD})=60^\circ
m
(
A
B
D
)
=
6
0
∘
,
m
(
B
D
C
^
)
=
4
5
∘
m(\widehat{BDC})=45^\circ
m
(
B
D
C
)
=
4
5
∘
, and
∣
B
D
∣
=
13
+
3
3
|BD|=13+3\sqrt 3
∣
B
D
∣
=
13
+
3
3
. What is
∣
A
C
∣
|AC|
∣
A
C
∣
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
20
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
18
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
16
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
14
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
12
<span class='latex-bold'>(A)</span>\ 20 \qquad<span class='latex-bold'>(B)</span>\ 18 \qquad<span class='latex-bold'>(C)</span>\ 16 \qquad<span class='latex-bold'>(D)</span>\ 14 \qquad<span class='latex-bold'>(E)</span>\ 12
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
20
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
18
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
16
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
14
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
12
geometry
trigonometry