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IberoAmerican
2019 IberoAmerican
3
3
Part of
2019 IberoAmerican
Problems
(1)
2019 Iberoamerican Mathematical Olympiad, P3
Source:
9/15/2019
Let
Γ
\Gamma
Γ
be the circumcircle of triangle
A
B
C
ABC
A
BC
. The line parallel to
A
C
AC
A
C
passing through
B
B
B
meets
Γ
\Gamma
Γ
at
D
D
D
(
D
≠
B
D\neq B
D
=
B
), and the line parallel to
A
B
AB
A
B
passing through
C
C
C
intersects
Γ
\Gamma
Γ
to
E
E
E
(
E
≠
C
E\neq C
E
=
C
). Lines
A
B
AB
A
B
and
C
D
CD
C
D
meet at
P
P
P
, and lines
A
C
AC
A
C
and
B
E
BE
BE
meet at
Q
Q
Q
. Let
M
M
M
be the midpoint of
D
E
DE
D
E
. Line
A
M
AM
A
M
meets
Γ
\Gamma
Γ
at
Y
Y
Y
(
Y
≠
A
Y\neq A
Y
=
A
) and line
P
Q
PQ
PQ
at
J
J
J
. Line
P
Q
PQ
PQ
intersects the circumcircle of triangle
B
C
J
BCJ
BC
J
at
Z
Z
Z
(
Z
≠
J
Z\neq J
Z
=
J
). If lines
B
Q
BQ
BQ
and
C
P
CP
CP
meet each other at
X
X
X
, show that
X
X
X
lies on the line
Y
Z
YZ
Y
Z
.
geometry
circumcircle