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National and Regional Contests
Turkey Contests
National Olympiad First Round
2012 National Olympiad First Round
21
21
Part of
2012 National Olympiad First Round
Problems
(1)
Turkish NMO First Round - 2012 Problem - 21 {Geometry}
Source:
7/1/2012
The angle bisector of vertex
A
A
A
of
△
A
B
C
\triangle ABC
△
A
BC
cuts
[
B
C
]
[BC]
[
BC
]
at
D
D
D
. The circle passing through
A
A
A
and touching to
B
C
BC
BC
at
D
D
D
meets
[
A
B
]
[AB]
[
A
B
]
and
[
A
C
]
[AC]
[
A
C
]
at
P
P
P
and
Q
Q
Q
, respectively.
A
D
AD
A
D
and
P
Q
PQ
PQ
meet at
T
T
T
. If
∣
A
B
∣
=
5
,
∣
B
C
∣
=
6
,
∣
C
A
∣
=
7
|AB|=5, |BC|=6, |CA|=7
∣
A
B
∣
=
5
,
∣
BC
∣
=
6
,
∣
C
A
∣
=
7
, then
∣
A
T
∣
∣
T
D
∣
=
?
\frac{|AT|}{|TD|}=?
∣
T
D
∣
∣
A
T
∣
=
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
7
5
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
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
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
7
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
<span class='latex-bold'>(A)</span>\ \frac75 \qquad <span class='latex-bold'>(B)</span>\ 2 \qquad <span class='latex-bold'>(C)</span>\ 3 \qquad <span class='latex-bold'>(D)</span>\ \frac72 \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
>
5
7
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
2
7
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
4
geometry
angle bisector