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Cono Sur Olympiad
2021 Cono Sur Olympiad
6
6
Part of
2021 Cono Sur Olympiad
Problems
(1)
P6 Cono Sur 2021
Source: Cono Sur 2021 #6
12/1/2021
Let
A
B
C
ABC
A
BC
be a scalene triangle with circle
Γ
\Gamma
Γ
. Let
P
,
Q
,
R
,
S
P,Q,R,S
P
,
Q
,
R
,
S
distinct points on the
B
C
BC
BC
side, in that order, such that
∠
B
A
P
=
∠
C
A
S
\angle BAP = \angle CAS
∠
B
A
P
=
∠
C
A
S
and
∠
B
A
Q
=
∠
C
A
R
\angle BAQ = \angle CAR
∠
B
A
Q
=
∠
C
A
R
. Let
U
,
V
,
W
,
Z
U, V, W, Z
U
,
V
,
W
,
Z
be the intersections, distinct from
A
A
A
, of the
A
P
,
A
Q
,
A
R
AP, AQ, AR
A
P
,
A
Q
,
A
R
and
A
S
AS
A
S
with
Γ
\Gamma
Γ
, respectively. Let
X
=
U
Q
∩
S
W
X = UQ \cap SW
X
=
U
Q
∩
S
W
,
Y
=
P
V
∩
Z
R
Y = PV \cap ZR
Y
=
P
V
∩
ZR
,
T
=
U
R
∩
V
S
T = UR \cap VS
T
=
U
R
∩
V
S
and
K
=
P
W
∩
Z
Q
K = PW \cap ZQ
K
=
P
W
∩
ZQ
. Suppose that the points
M
M
M
and
N
N
N
are well determined, such that
M
=
K
X
∩
T
Y
M = KX \cap TY
M
=
K
X
∩
T
Y
and
N
=
T
X
∩
K
Y
N = TX \cap KY
N
=
TX
∩
K
Y
. Show that
M
,
N
,
A
M, N, A
M
,
N
,
A
are collinear.
geometry