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Vietnam Contests
Vietnam Team Selection Test
2023 Vietnam Team Selection Test
5
5
Part of
2023 Vietnam Team Selection Test
Problems
(1)
Contrived geo strikes again
Source: Vietnam TST 2023 P5
4/14/2023
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral with
∠
B
<
∠
A
<
9
0
o
\angle B < \angle A < 90^{o}
∠
B
<
∠
A
<
9
0
o
. Let
I
I
I
be the midpoint of
A
B
AB
A
B
and
S
S
S
the intersection of
A
D
AD
A
D
and
B
C
BC
BC
. Let
R
R
R
be a variable point inside the triangle
S
A
B
SAB
S
A
B
such that
∠
A
S
R
=
∠
B
S
R
\angle ASR = \angle BSR
∠
A
SR
=
∠
BSR
. On the straight lines
A
R
,
B
R
AR, BR
A
R
,
BR
, take the points
E
,
F
E, F
E
,
F
, respectively so that
B
E
,
A
F
BE , AF
BE
,
A
F
are parallel to
R
S
RS
RS
. Suppose that
E
F
EF
EF
intersects the circumcircle of triangle
S
A
B
SAB
S
A
B
at points
H
,
K
H, K
H
,
K
. On the segment
A
B
AB
A
B
, take points
M
,
N
M , N
M
,
N
such that
∠
A
H
M
=
∠
B
H
I
\angle AHM =\angle BHI
∠
A
H
M
=
∠
B
H
I
,
∠
B
K
N
=
∠
A
K
I
\angle BKN = \angle AKI
∠
B
K
N
=
∠
A
K
I
. a) Prove that the center
J
J
J
of the circumcircle of triangle
S
M
N
SMN
SMN
lies on a fixed line. b) On
B
E
,
A
F
BE, AF
BE
,
A
F
, take the points
P
,
Q
P, Q
P
,
Q
respectively so that
C
P
CP
CP
is parallel to
S
E
SE
SE
and
D
Q
DQ
D
Q
is parallel to
S
F
SF
SF
. The lines
S
E
,
S
F
SE, SF
SE
,
SF
intersect the circle
(
S
A
B
)
(SAB)
(
S
A
B
)
, respectively, at
U
,
V
U, V
U
,
V
. Let
G
G
G
be the intersection of
A
U
AU
A
U
and
B
V
BV
B
V
. Prove that the median of vertex
G
G
G
of the triangle
G
P
Q
GPQ
GPQ
always passes through a fixed point .
geometry