MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2014 National Olympiad First Round
1
1
Part of
2014 National Olympiad First Round
Problems
(1)
P01 [Geometry] - Turkish NMO 1st Round - 2014
Source:
5/21/2014
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral such that
m
(
D
A
B
^
)
=
m
(
C
B
D
^
)
=
12
0
∘
m \left (\widehat{DAB} \right )=m \left (\widehat{CBD} \right )=120^{\circ}
m
(
D
A
B
)
=
m
(
CB
D
)
=
12
0
∘
,
∣
A
B
∣
=
2
|AB|=2
∣
A
B
∣
=
2
,
∣
A
D
∣
=
4
|AD|=4
∣
A
D
∣
=
4
and
∣
B
C
∣
=
∣
B
D
∣
|BC|=|BD|
∣
BC
∣
=
∣
B
D
∣
. If the line through
C
C
C
which is parallel to
A
B
AB
A
B
meets
A
D
AD
A
D
at
E
E
E
, what is
∣
C
E
∣
|CE|
∣
CE
∣
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
8
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
7
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
6
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
5
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
None of the preceding
<span class='latex-bold'>(A)</span>\ 8 \qquad<span class='latex-bold'>(B)</span>\ 7 \qquad<span class='latex-bold'>(C)</span>\ 6 \qquad<span class='latex-bold'>(D)</span>\ 5 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the preceding}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
8
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
7
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
6
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the preceding
geometry