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National and Regional Contests
Singapore Contests
Singapore MO Open
2023 Sinapore MO Open
2023 Sinapore MO Open
Part of
Singapore MO Open
Subcontests
(1)
P1
1
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Conditional geo with centroid
In a scalene triangle
A
B
C
ABC
A
BC
with centroid
G
G
G
and circumcircle
ω
\omega
ω
centred at
O
O
O
, the extension of
A
G
AG
A
G
meets
ω
\omega
ω
at
M
M
M
; lines
A
B
AB
A
B
and
C
M
CM
CM
intersect at
P
P
P
; and lines
A
C
AC
A
C
and
B
M
BM
BM
intersect at
Q
Q
Q
. Suppose the circumcentre
S
S
S
of the triangle
A
P
Q
APQ
A
PQ
lies on
ω
\omega
ω
and
A
,
O
,
S
A, O, S
A
,
O
,
S
are collinear. Prove that
∠
A
G
O
=
9
0
o
\angle AGO = 90^{o}
∠
A
GO
=
9
0
o
.