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International Contests
APMO
2024 APMO
5
5
Part of
2024 APMO
Problems
(1)
Miquel spam geo
Source: APMO 2024 P5
7/29/2024
Line
ℓ
\ell
ℓ
intersects sides
B
C
BC
BC
and
A
D
AD
A
D
of cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
in its interior points
R
R
R
and
S
S
S
, respectively, and intersects ray
D
C
DC
D
C
beyond point
C
C
C
at
Q
Q
Q
, and ray
B
A
BA
B
A
beyond point
A
A
A
at
P
P
P
. Circumcircles of the triangles
Q
C
R
QCR
QCR
and
Q
D
S
QDS
Q
D
S
intersect at
N
≠
Q
N \neq Q
N
=
Q
, while circumcircles of the triangles
P
A
S
PAS
P
A
S
and
P
B
R
PBR
PBR
intersect at
M
≠
P
M\neq P
M
=
P
. Let lines
M
P
MP
MP
and
N
Q
NQ
NQ
meet at point
X
X
X
, lines
A
B
AB
A
B
and
C
D
CD
C
D
meet at point
K
K
K
and lines
B
C
BC
BC
and
A
D
AD
A
D
meet at point
L
L
L
. Prove that point
X
X
X
lies on line
K
L
KL
K
L
.
geometry
APMO 2024