MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2007 National Olympiad First Round
17
17
Part of
2007 National Olympiad First Round
Problems
(1)
Turkey NMO 2007 1st Round - P17 (Geometry)
Source:
10/4/2012
Let
K
K
K
be the point of intersection of
A
B
AB
A
B
and the line touching the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
at
C
C
C
where
m
(
A
^
)
>
m
(
B
^
)
m(\widehat {A}) > m(\widehat {B})
m
(
A
)
>
m
(
B
)
. Let
L
L
L
be a point on
[
B
C
]
[BC]
[
BC
]
such that
m
(
A
L
B
^
)
=
m
(
C
A
K
^
)
m(\widehat{ALB})=m(\widehat{CAK})
m
(
A
L
B
)
=
m
(
C
A
K
)
,
5
∣
L
C
∣
=
4
∣
B
L
∣
5|LC|=4|BL|
5∣
L
C
∣
=
4∣
B
L
∣
, and
∣
K
C
∣
=
12
|KC|=12
∣
K
C
∣
=
12
. What is
∣
A
K
∣
|AK|
∣
A
K
∣
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
4
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
6
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
8
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
9
<
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 above
<span class='latex-bold'>(A)</span>\ 4\sqrt 2 \qquad<span class='latex-bold'>(B)</span>\ 6 \qquad<span class='latex-bold'>(C)</span>\ 8 \qquad<span class='latex-bold'>(D)</span>\ 9 \qquad<span class='latex-bold'>(E)</span>\ \text{None of the above}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
4
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
6
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
8
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
9
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None of the above
geometry
circumcircle