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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2023 Vietnam National Olympiad
4
4
Part of
2023 Vietnam National Olympiad
Problems
(1)
Vietnamese geo with way too many points
Source: VMO 2023 day 1 P4
2/24/2023
Given is a triangle
A
B
C
ABC
A
BC
and let
D
D
D
be the midpoint the major arc
B
A
C
BAC
B
A
C
of its circumcircle. Let
M
,
N
M , N
M
,
N
be the midpoints of
A
B
,
A
C
AB , AC
A
B
,
A
C
and
J
,
E
,
F
J , E , F
J
,
E
,
F
are the touchpoints of the incircle
(
I
)
(I)
(
I
)
of
△
A
B
C
\triangle ABC
△
A
BC
with
B
C
,
C
A
,
A
B
BC, CA, AB
BC
,
C
A
,
A
B
. The line
M
N
MN
MN
intersects
J
E
,
J
F
JE , JF
J
E
,
J
F
at
K
,
H
K , H
K
,
H
respectively;
I
J
IJ
I
J
intersects the circle
(
B
I
C
)
(BIC)
(
B
I
C
)
at
G
G
G
and
D
G
DG
D
G
intersects
(
B
I
C
)
(BIC)
(
B
I
C
)
at
T
T
T
. a) Prove that
J
A
JA
J
A
passes through the midpoint of
H
K
HK
HK
and is perpendicular to
I
T
IT
I
T
. b) Let
R
,
S
R, S
R
,
S
respectively be the perpendicular projection of
D
D
D
on
A
B
,
A
C
AB, AC
A
B
,
A
C
. Take the points
P
,
Q
P, Q
P
,
Q
on
I
F
,
I
E
IF , IE
I
F
,
I
E
respectively such that
K
P
KP
K
P
and
H
Q
HQ
H
Q
are both perpendicular to
M
N
MN
MN
. Prove that the three lines
M
P
,
N
Q
MP , NQ
MP
,
NQ
and
R
S
RS
RS
are concurrent .
geometry