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National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2018 Vietnam National Olympiad
7
7
Part of
2018 Vietnam National Olympiad
Problems
(1)
VMO 2018 P7
Source: Vietnam MO 2nd day 3rd problem (last problem)
1/12/2018
Acute scalene triangle
A
B
C
ABC
A
BC
has
G
G
G
as its centroid and
O
O
O
as its circumcenter. Let
H
a
,
H
b
,
H
c
H_a,\, H_b,\, H_c
H
a
,
H
b
,
H
c
be the projections of
A
,
B
,
C
A,\, B,\, C
A
,
B
,
C
on respective opposite sides and
D
,
E
,
F
D,\, E,\, F
D
,
E
,
F
be the midpoints of
B
C
,
C
A
,
A
B
BC,\, CA,\, AB
BC
,
C
A
,
A
B
in that order.
G
H
a
→
,
G
H
b
→
,
G
H
c
→
\overrightarrow{GH_a},\, \overrightarrow{GH_b},\, \overrightarrow{GH_c}
G
H
a
,
G
H
b
,
G
H
c
intersect
(
O
)
(O)
(
O
)
at
X
,
Y
,
Z
X,\,Y,\,Z
X
,
Y
,
Z
respectively. a. Prove that the circle
(
X
C
E
)
(XCE)
(
XCE
)
pass through the midpoint of
B
H
a
BH_a
B
H
a
b. Let
M
,
N
,
P
M,\, N,\, P
M
,
N
,
P
be the midpoints of
A
X
,
B
Y
,
C
Z
AX,\, BY,\, CZ
A
X
,
B
Y
,
CZ
respectively. Prove that
D
M
↔
,
E
N
↔
,
F
P
↔
\overleftrightarrow{DM},\, \overleftrightarrow{EN},\,\overleftrightarrow{FP}
D
M
,
EN
,
FP
are concurrent.
geometry