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Balkan MO Shortlist
2022 Balkan MO Shortlist
G6
G6
Part of
2022 Balkan MO Shortlist
Problems
(1)
Bash geo
Source: BMO Shortlist 2022, G6 & Romanian TST 2022, Day 3 P2
5/13/2023
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
<
A
C
AB < AC
A
B
<
A
C
and let
D
D{}
D
be the other intersection point of the angle bisector of
∠
A
\angle A
∠
A
with the circumcircle of the triangle
A
B
C
ABC
A
BC
. Let
E
E{}
E
and
F
F{}
F
be points on the sides
A
B
AB
A
B
and
A
C
AC
A
C
respectively, such that
A
E
=
A
F
AE = AF
A
E
=
A
F
and let
P
P{}
P
be the point of intersection of
A
D
AD
A
D
and
E
F
EF
EF
. Let
M
M{}
M
be the midpoint of
B
C
BC{}
BC
. Prove that
A
M
AM
A
M
and the circumcircles of the triangles
A
E
F
AEF
A
EF
and
P
M
D
PMD
PM
D
pass through a common point.
geometry