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Rioplatense Mathematical Olympiad, Level 3
2019 Rioplatense Mathematical Olympiad, Level 3
5
5
Part of
2019 Rioplatense Mathematical Olympiad, Level 3
Problems
(1)
Rio Geo.
Source: Rioplatense Olympiad L3 2019
12/10/2019
Let
A
B
C
ABC
A
BC
be a triangle with
A
B
<
A
C
AB<AC
A
B
<
A
C
and circuncircle
ω
\omega
ω
. Let
M
M
M
and
N
N
N
be the midpoints of
A
C
AC
A
C
and
A
B
AB
A
B
respectively and
G
G
G
is the centroid of
A
B
C
ABC
A
BC
. Let
P
P
P
be the foot of perpendicular of
A
A
A
to the line
B
C
BC
BC
, and the point
Q
Q
Q
is the intersection of
G
P
GP
GP
and
ω
\omega
ω
(
Q
,
P
,
G
Q,P,G
Q
,
P
,
G
are collinears in this order). The line
Q
M
QM
QM
cuts
ω
\omega
ω
in
M
1
M_1
M
1
and the line
Q
N
QN
QN
cuts
ω
\omega
ω
in
N
1
N_1
N
1
. If
K
K
K
is the intersection of
B
M
1
BM_1
B
M
1
and
C
N
1
CN_1
C
N
1
prove that
P
P
P
,
G
G
G
and
K
K
K
are collinears.
geometry