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International Contests
Middle European Mathematical Olympiad
2017 Middle European Mathematical Olympiad
6
6
Part of
2017 Middle European Mathematical Olympiad
Problems
(1)
Fairly standard configuration involving tangents
Source: MEMO 2017 T6
8/25/2017
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with
A
B
≠
A
C
AB \neq AC
A
B
=
A
C
, circumcentre
O
O
O
and circumcircle
Γ
\Gamma
Γ
. Let the tangents to
Γ
\Gamma
Γ
at
B
B
B
and
C
C
C
meet each other at
D
D
D
, and let the line
A
O
AO
A
O
intersect
B
C
BC
BC
at
E
E
E
. Denote the midpoint of
B
C
BC
BC
by
M
M
M
and let
A
M
AM
A
M
meet
Γ
\Gamma
Γ
again at
N
≠
A
N \neq A
N
=
A
. Finally, let
F
≠
A
F \neq A
F
=
A
be a point on
Γ
\Gamma
Γ
such that
A
,
M
,
E
A, M, E
A
,
M
,
E
and
F
F
F
are concyclic. Prove that
F
N
FN
FN
bisects the segment
M
D
MD
M
D
.
geometry