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Balkan MO Shortlist
2022 Balkan MO Shortlist
G3
G3
Part of
2022 Balkan MO Shortlist
Problems
(1)
Long geo
Source: BMO Shortlist 2022, G3
5/13/2023
Let
A
B
C
ABC
A
BC
a triangle and let
ω
\omega
ω
be its circumcircle. Let
E
E{}
E
be the midpoint of the minor arc
B
C
BC
BC
of
ω
\omega
ω
, and
M
M{}
M
the midpoint of
B
C
BC
BC
. Let
V
V
V
be the other point of intersection of
A
M
AM
A
M
with
ω
\omega
ω
,
F
F{}
F
the point of intersection of
A
E
AE
A
E
with
B
C
BC
BC
,
X
X{}
X
the other point of intersection of the circumcircle of
F
E
M
FEM
FEM
with
ω
\omega
ω
,
X
′
X'
X
′
the reflection of
V
V{}
V
with respect to
M
M{}
M
,
A
′
A'{}
A
′
the foot of the perpendicular from
A
A{}
A
to
B
C
BC
BC
and
S
S{}
S
the other point of intersection of
X
A
′
XA'
X
A
′
with
ω
\omega
ω
. If
Z
∈
ω
Z \in \omega
Z
∈
ω
with
Z
≠
X
Z\neq X
Z
=
X
is such that
A
X
=
A
Z
AX = AZ
A
X
=
A
Z
, then prove that
S
,
X
′
S, X'
S
,
X
′
and
Z
Z{}
Z
are collinear.
geometry