MathDB
Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2014 China Second Round Olympiad
2
2
Part of
2014 China Second Round Olympiad
Problems
(1)
2014 China Second Round Olympiad Second Part Problem 2
Source: 2014 China Second Round Olympiad
8/4/2015
Let
A
B
C
ABC
A
BC
be an acute triangle such that
∠
B
A
C
≠
6
0
∘
\angle BAC \neq 60^\circ
∠
B
A
C
=
6
0
∘
. Let
D
,
E
D,E
D
,
E
be points such that
B
D
,
C
E
BD,CE
B
D
,
CE
are tangent to the circumcircle of
A
B
C
ABC
A
BC
and
B
D
=
C
E
=
B
C
BD=CE=BC
B
D
=
CE
=
BC
(
A
A
A
is on one side of line
B
C
BC
BC
and
D
,
E
D,E
D
,
E
are on the other side). Let
F
,
G
F,G
F
,
G
be intersections of line
D
E
DE
D
E
and lines
A
B
,
A
C
AB,AC
A
B
,
A
C
. Let
M
M
M
be intersection of
C
F
CF
CF
and
B
D
BD
B
D
, and
N
N
N
be intersection of
C
E
CE
CE
and
B
G
BG
BG
. Prove that
A
M
=
A
N
AM=AN
A
M
=
A
N
.
geometry
circumcircle