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APMO
2024 APMO
1
1
Part of
2024 APMO
Problems
(1)
A lies on the radical axis of BQX and CPX
Source: APMO 2024 P1
7/29/2024
Let
A
B
C
ABC
A
BC
be an acute triangle. Let
D
D
D
be a point on side
A
B
AB
A
B
and
E
E
E
be a point on side
A
C
AC
A
C
such that lines
B
C
BC
BC
and
D
E
DE
D
E
are parallel. Let
X
X
X
be an interior point of
B
C
E
D
BCED
BCE
D
. Suppose rays
D
X
DX
D
X
and
E
X
EX
EX
meet side
B
C
BC
BC
at points
P
P
P
and
Q
Q
Q
, respectively, such that both
P
P
P
and
Q
Q
Q
lie between
B
B
B
and
C
C
C
. Suppose that the circumcircles of triangles
B
Q
X
BQX
BQX
and
C
P
X
CPX
CPX
intersect at a point
Y
≠
X
Y \neq X
Y
=
X
. Prove that the points
A
,
X
A, X
A
,
X
, and
Y
Y
Y
are collinear.
geometry
APMO