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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2018 Vietnam National Olympiad
2
2
Part of
2018 Vietnam National Olympiad
Problems
(1)
VMO 2018 P2
Source: Vietnam Mo 2018 1st day 2nd problem
1/11/2018
We have a scalene acute triangle
A
B
C
ABC
A
BC
(triangle with no two equal sides) and a point
D
D
D
on side
B
C
BC
BC
. Pick a point
E
E
E
on side
A
B
AB
A
B
and a point
F
F
F
on side
A
C
AC
A
C
such that
∠
D
E
B
=
∠
D
F
C
\angle DEB=\angle DFC
∠
D
EB
=
∠
D
FC
. Lines
D
F
,
D
E
DF,\, DE
D
F
,
D
E
intersect
A
B
,
A
C
AB,\, AC
A
B
,
A
C
at points
M
,
N
M,\, N
M
,
N
, respectively. Denote
(
I
1
)
,
(
I
2
)
(I_1),\, (I_2)
(
I
1
)
,
(
I
2
)
by the circumcircles of triangles
D
E
M
,
D
F
N
DEM,\, DFN
D
EM
,
D
FN
in that order. The circle
(
J
1
)
(J_1)
(
J
1
)
touches
(
I
1
)
(I_1)
(
I
1
)
internally at
D
D
D
and touches
A
B
AB
A
B
at
K
K
K
, circle
(
J
2
)
(J_2)
(
J
2
)
touches
(
I
2
)
(I_2)
(
I
2
)
internally at
D
D
D
and touches
A
C
AC
A
C
at
H
H
H
.
P
P
P
is the intersection of
(
I
1
)
,
(
I
2
)
(I_1),\, (I_2)
(
I
1
)
,
(
I
2
)
different from
D
D
D
.
Q
Q
Q
is the intersection of
(
J
1
)
,
(
J
2
)
(J_1),\, (J_2)
(
J
1
)
,
(
J
2
)
different from
D
D
D
. a. Prove that all points
D
,
P
,
Q
D,\, P,\, Q
D
,
P
,
Q
lie on the same line. b. The circumcircles of triangles
A
E
F
,
A
H
K
AEF,\, AHK
A
EF
,
A
HK
intersect at
A
,
G
A,\, G
A
,
G
.
(
A
E
F
)
(AEF)
(
A
EF
)
also cut
A
Q
AQ
A
Q
at
A
,
L
A,\, L
A
,
L
. Prove that the tangent at
D
D
D
of
(
D
Q
G
)
(DQG)
(
D
QG
)
cuts
E
F
EF
EF
at a point on
(
D
L
G
)
(DLG)
(
D
L
G
)
.
geometry