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2019 JBMO Shortlist
G2
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Part of
2019 JBMO Shortlist
Problems
(1)
Geometry
Source: 8th European Mathematical Cup 2019 Junior Q3
12/26/2019
Let
A
B
C
ABC
A
BC
be a triangle with circumcircle
ω
\omega
ω
. Let
l
B
l_B
l
B
and
l
C
l_C
l
C
be two lines through the points
B
B
B
and
C
C
C
, respectively, such that
l
B
∥
l
C
l_B \parallel l_C
l
B
∥
l
C
. The second intersections of
l
B
l_B
l
B
and
l
C
l_C
l
C
with
ω
\omega
ω
are
D
D
D
and
E
E
E
, respectively. Assume that
D
D
D
and
E
E
E
are on the same side of
B
C
BC
BC
as
A
A
A
. Let
D
A
DA
D
A
intersect
l
C
l_C
l
C
at
F
F
F
and let
E
A
EA
E
A
intersect
l
B
l_B
l
B
at
G
G
G
. If
O
O
O
,
O
1
O_1
O
1
and
O
2
O_2
O
2
are circumcenters of the triangles
A
B
C
ABC
A
BC
,
A
D
G
ADG
A
D
G
and
A
E
F
AEF
A
EF
, respectively, and
P
P
P
is the circumcenter of the triangle
O
O
1
O
2
OO_1O_2
O
O
1
O
2
, prove that
l
B
∥
O
P
∥
l
C
l_B \parallel OP \parallel l_C
l
B
∥
OP
∥
l
C
.Proposed by Stefan Lozanovski, Macedonia
geometry