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Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2008 National Olympiad First Round
1
1
Part of
2008 National Olympiad First Round
Problems
(1)
Turkey NMO 2008 1st Round - P01 (Geometry)
Source:
8/23/2012
Let
A
D
AD
A
D
be a median of
△
A
B
C
\triangle ABC
△
A
BC
such that
m
(
A
D
B
^
)
=
4
5
∘
m(\widehat{ADB})=45^{\circ}
m
(
A
D
B
)
=
4
5
∘
and
m
(
A
C
B
^
)
=
3
0
∘
m(\widehat{ACB})=30^{\circ}
m
(
A
CB
)
=
3
0
∘
. What is the measure of
A
B
C
^
\widehat{ABC}
A
BC
in degrees?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
75
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
90
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
105
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
120
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
135
<span class='latex-bold'>(A)</span>\ 75 \qquad<span class='latex-bold'>(B)</span>\ 90 \qquad<span class='latex-bold'>(C)</span>\ 105 \qquad<span class='latex-bold'>(D)</span>\ 120 \qquad<span class='latex-bold'>(E)</span>\ 135
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
75
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
90
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
105
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
120
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
135
geometry
circumcircle