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Junior Balkan MO
2017 Junior Balkan MO
3
3
Part of
2017 Junior Balkan MO
Problems
(1)
A,M,X are collinear
Source: JBMO 2017, Q3
6/26/2017
Let
A
B
C
ABC
A
BC
be an acute triangle such that
A
B
≠
A
C
AB\neq AC
A
B
=
A
C
,with circumcircle
Γ
\Gamma
Γ
and circumcenter
O
O
O
. Let
M
M
M
be the midpoint of
B
C
BC
BC
and
D
D
D
be a point on
Γ
\Gamma
Γ
such that
A
D
⊥
B
C
AD \perp BC
A
D
⊥
BC
. let
T
T
T
be a point such that
B
D
C
T
BDCT
B
D
CT
is a parallelogram and
Q
Q
Q
a point on the same side of
B
C
BC
BC
as
A
A
A
such that
∠
B
Q
M
=
∠
B
C
A
\angle{BQM}=\angle{BCA}
∠
BQM
=
∠
BC
A
and
∠
C
Q
M
=
∠
C
B
A
\angle{CQM}=\angle{CBA}
∠
CQM
=
∠
CB
A
. Let the line
A
O
AO
A
O
intersect
Γ
\Gamma
Γ
at
E
E
E
(
E
≠
A
)
(E\neq A)
(
E
=
A
)
and let the circumcircle of
△
E
T
Q
\triangle ETQ
△
ETQ
intersect
Γ
\Gamma
Γ
at point
X
≠
E
X\neq E
X
=
E
. Prove that the point
A
,
M
A,M
A
,
M
and
X
X
X
are collinear.
geometry