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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2021 Vietnam National Olympiad
3
3
Part of
2021 Vietnam National Olympiad
Problems
(1)
Merry Christmas
Source: VMO 2021 P3
12/25/2020
Let
△
A
B
C
\bigtriangleup ABC
△
A
BC
is not an isosceles triangle and is an acute triangle,
A
D
,
B
E
,
C
F
AD,BE,CF
A
D
,
BE
,
CF
be the altitudes and
H
H
H
is the orthocenter .Let
I
I
I
is the circumcenter of
△
H
E
F
\bigtriangleup HEF
△
H
EF
and let
K
,
J
K,J
K
,
J
is the midpoint of
B
C
,
E
F
BC,EF
BC
,
EF
respectively.Let
H
J
HJ
H
J
intersects
(
I
)
(I)
(
I
)
again at
G
G
G
and
G
K
GK
G
K
intersects
(
I
)
(I)
(
I
)
at
L
≠
G
L\neq G
L
=
G
. a) Prove that
A
L
AL
A
L
is perpendicular to
E
F
EF
EF
. b) Let
A
L
AL
A
L
intersects
E
F
EF
EF
at
M
M
M
, the line
I
M
IM
I
M
intersects the circumcircle
△
I
E
F
\bigtriangleup IEF
△
I
EF
again at
N
N
N
,
D
N
DN
D
N
intersects
A
B
,
A
C
AB,AC
A
B
,
A
C
at
P
P
P
and
Q
Q
Q
respectively then prove that
P
E
,
Q
F
,
A
K
PE,QF,AK
PE
,
QF
,
A
K
are concurrent.
geometry
circumcircle