MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2005 National Olympiad First Round
17
17
Part of
2005 National Olympiad First Round
Problems
(1)
P17 [Geometry] - Turkish NMO 1st Round - 2005
Source:
10/26/2013
Construct outer squares
A
B
M
N
ABMN
A
BMN
,
B
C
K
L
BCKL
BC
K
L
,
A
C
P
Q
ACPQ
A
CPQ
on sides
[
A
B
]
[AB]
[
A
B
]
,
[
B
C
]
[BC]
[
BC
]
,
[
C
A
]
[CA]
[
C
A
]
of triangle
A
B
C
ABC
A
BC
, respectively. Construct squares
N
Q
Z
T
NQZT
NQZT
and
K
P
Y
X
KPYX
K
P
Y
X
on segments
[
N
Q
]
[NQ]
[
NQ
]
and
[
K
P
]
[KP]
[
K
P
]
. If
A
r
e
a
(
A
B
M
N
)
−
A
r
e
a
(
B
C
K
L
)
=
1
Area(ABMN) - Area(BCKL)=1
A
re
a
(
A
BMN
)
−
A
re
a
(
BC
K
L
)
=
1
, what is
A
r
e
a
(
N
Q
Z
T
)
−
A
r
e
a
(
K
P
Y
X
)
Area(NQZT)-Area(KPYX)
A
re
a
(
NQZT
)
−
A
re
a
(
K
P
Y
X
)
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
3
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
5
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
4
<span class='latex-bold'>(A)</span>\ \dfrac 34 \qquad<span class='latex-bold'>(B)</span>\ \dfrac 53 \qquad<span class='latex-bold'>(C)</span>\ 2 \qquad<span class='latex-bold'>(D)</span>\ 3 \qquad<span class='latex-bold'>(E)</span>\ 4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
4
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
3
5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
4
geometry
trigonometry