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Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2006 National Olympiad First Round
21
21
Part of
2006 National Olympiad First Round
Problems
(1)
Turkey NMO 2006 1st Round - P21 (Geometry)
Source:
2/2/2013
Let
A
B
C
ABC
A
BC
be a triangle with
m
(
A
^
)
=
7
0
∘
m(\widehat A) = 70^\circ
m
(
A
)
=
7
0
∘
and the incenter
I
I
I
. If
∣
B
C
∣
=
∣
A
C
∣
+
∣
A
I
∣
|BC|=|AC|+|AI|
∣
BC
∣
=
∣
A
C
∣
+
∣
A
I
∣
, then what is
m
(
B
^
)
m(\widehat B)
m
(
B
)
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
3
5
∘
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
3
6
∘
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
4
2
∘
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
4
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 above
<span class='latex-bold'>(A)</span>\ 35^\circ \qquad<span class='latex-bold'>(B)</span>\ 36^\circ \qquad<span class='latex-bold'>(C)</span>\ 42^\circ \qquad<span class='latex-bold'>(D)</span>\ 45^\circ \qquad<span class='latex-bold'>(E)</span>\ \text{None of above}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
3
5
∘
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
3
6
∘
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
4
2
∘
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
4
5
∘
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of above
geometry
incenter