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Turkey MO (2nd round)
2018 Turkey MO (2nd Round)
2
2
Part of
2018 Turkey MO (2nd Round)
Problems
(1)
Triangle
Source: Turkey National Mathematical Olympiad 2018
12/2/2018
Let
P
P
P
be a point in the interior of the triangle
A
B
C
ABC
A
BC
. The lines
A
P
AP
A
P
,
B
P
BP
BP
, and
C
P
CP
CP
intersect the sides
B
C
BC
BC
,
C
A
CA
C
A
, and
A
B
AB
A
B
at
D
,
E
D,E
D
,
E
, and
F
F
F
, respectively. A point
Q
Q
Q
is taken on the ray
[
B
E
[BE
[
BE
such that
E
∈
[
B
Q
]
E\in [BQ]
E
∈
[
BQ
]
and
m
(
E
D
Q
^
)
=
m
(
B
D
F
^
)
m(\widehat{EDQ})=m(\widehat{BDF})
m
(
E
D
Q
)
=
m
(
B
D
F
)
. If
B
E
BE
BE
and
A
D
AD
A
D
are perpendicular, and
∣
D
Q
∣
=
2
∣
B
D
∣
|DQ|=2|BD|
∣
D
Q
∣
=
2∣
B
D
∣
, prove that
m
(
F
D
E
^
)
=
6
0
∘
m(\widehat{FDE})=60^\circ
m
(
F
D
E
)
=
6
0
∘
.
geometry