MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
1999 National Olympiad First Round
1
1
Part of
1999 National Olympiad First Round
Problems
(1)
Turkish NMO First Round - 1999 P-01 (Geometry)
Source:
7/3/2012
Let
A
B
C
ABC
A
BC
be a triangle with \left|AB\right| \equal{} 14, \left|BC\right| \equal{} 12, \left|AC\right| \equal{} 10. Let
D
D
D
be a point on
[
A
C
]
\left[AC\right]
[
A
C
]
and
E
E
E
be a point on
[
B
C
]
\left[BC\right]
[
BC
]
such that \left|AD\right| \equal{} 4 and Area\left(ABC\right) \equal{} 2Area\left(CDE\right). Find
A
r
e
a
(
A
B
E
)
Area\left(ABE\right)
A
re
a
(
A
BE
)
.
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
4
6
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
6
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
3
6
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
4
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
4
5
<span class='latex-bold'>(A)</span>\ 4\sqrt {6} \qquad<span class='latex-bold'>(B)</span>\ 6\sqrt {2} \qquad<span class='latex-bold'>(C)</span>\ 3\sqrt {6} \qquad<span class='latex-bold'>(D)</span>\ 4\sqrt {2} \qquad<span class='latex-bold'>(E)</span>\ 4\sqrt {5}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
4
6
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
6
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
3
6
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
4
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
4
5
geometry