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National and Regional Contests
Turkey Contests
National Olympiad First Round
2012 National Olympiad First Round
9
9
Part of
2012 National Olympiad First Round
Problems
(1)
Turkish NMO First Round - 2012 Problem - 09 {Geometry}
Source:
7/1/2012
The chord
[
C
D
]
[CD]
[
C
D
]
of the circle with diameter
[
A
B
]
[AB]
[
A
B
]
is perpendicular to
[
A
B
]
[AB]
[
A
B
]
. Let
M
M
M
and
N
N
N
be the midpoints of
[
B
C
]
[BC]
[
BC
]
and
[
A
D
]
[AD]
[
A
D
]
, respectively. If
∣
B
C
∣
=
6
|BC|=6
∣
BC
∣
=
6
and
∣
A
D
∣
=
2
3
|AD|=2\sqrt{3}
∣
A
D
∣
=
2
3
, then
∣
M
N
∣
=
?
|MN|=?
∣
MN
∣
=
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
3
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
21
<
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
<span class='latex-bold'>(A)</span>\ 4 \qquad <span class='latex-bold'>(B)</span>\ 3 \sqrt 2 \qquad <span class='latex-bold'>(C)</span>\ \sqrt{21} \qquad <span class='latex-bold'>(D)</span>\ 5 \qquad <span class='latex-bold'>(E)</span>\ \text{None}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
4
<
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
>
21
<
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
geometry