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National and Regional Contests
Korea Contests
Korea Junior Mathematics Olympiad
2016 Korea Junior Math Olympiad
6
6
Part of
2016 Korea Junior Math Olympiad
Problems
(1)
nice geometry
Source: 2016 KJMO #6
11/13/2016
circle
O
1
O_1
O
1
is tangent to
A
C
AC
A
C
,
B
C
BC
BC
(side of triangle
A
B
C
ABC
A
BC
) at point
D
,
E
D, E
D
,
E
. circle
O
2
O_2
O
2
include
O
1
O_1
O
1
, is tangent to
B
C
BC
BC
,
A
B
AB
A
B
(side of triangle
A
B
C
ABC
A
BC
) at point
E
,
F
E, F
E
,
F
The tangent of
O
2
O_2
O
2
at
P
(
D
E
∩
O
2
,
P
≠
E
)
P(DE \cap O_2, P \neq E)
P
(
D
E
∩
O
2
,
P
=
E
)
meets
A
B
AB
A
B
at
Q
Q
Q
. A line passing through
O
1
O_1
O
1
(center of
O
1
O_1
O
1
) and parallel to
B
O
2
BO_2
B
O
2
(
O
2
O_2
O
2
is also center of
O
2
O_2
O
2
) meets
B
C
BC
BC
at
G
G
G
,
E
Q
∩
A
C
=
K
,
K
G
∩
E
F
=
L
EQ \cap AC=K, KG \cap EF=L
EQ
∩
A
C
=
K
,
K
G
∩
EF
=
L
,
E
O
2
EO_2
E
O
2
meets circle
O
2
O_2
O
2
at
N
(
≠
E
)
N(\neq E)
N
(
=
E
)
,
L
O
2
∩
F
N
=
M
LO_2 \cap FN=M
L
O
2
∩
FN
=
M
. IF
N
N
N
is a middle point of
F
M
FM
FM
, prove that
B
G
=
2
E
G
BG=2EG
BG
=
2
EG
geometry