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Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2005 National Olympiad First Round
13
13
Part of
2005 National Olympiad First Round
Problems
(1)
P13 [Geometry] - Turkish NMO 1st Round - 2005
Source:
10/26/2013
Let
A
B
C
D
ABCD
A
BC
D
be an isosceles trapezoid such that its diagonal is
3
\sqrt 3
3
and its base angle is
6
0
∘
60^\circ
6
0
∘
, where
A
D
∥
B
C
AD \parallel BC
A
D
∥
BC
. Let
P
P
P
be a point on the plane of the trapezoid such that
∣
P
A
∣
=
1
|PA|=1
∣
P
A
∣
=
1
and
∣
P
D
∣
=
3
|PD|=3
∣
P
D
∣
=
3
. Which of the following can be the length of
[
P
C
]
[PC]
[
PC
]
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
6
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
2
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
2
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
3
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
7
<span class='latex-bold'>(A)</span>\ \sqrt 6 \qquad<span class='latex-bold'>(B)</span>\ 2\sqrt 2 \qquad<span class='latex-bold'>(C)</span>\ 2 \sqrt 3 \qquad<span class='latex-bold'>(D)</span>\ 3\sqrt 3 \qquad<span class='latex-bold'>(E)</span>\ \sqrt 7
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
6
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
2
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
2
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
3
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
7
geometry
trapezoid