MathDB
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Contests
National and Regional Contests
India Contests
India National Olympiad
2018 India National Olympiad
3
3
Part of
2018 India National Olympiad
Problems
(1)
INMO 2018 -- Problem #3
Source: INMO 2018
1/21/2018
Let
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
be two circles with respective centres
O
1
O_1
O
1
and
O
2
O_2
O
2
intersecting in two distinct points
A
A
A
and
B
B
B
such that
∠
O
1
A
O
2
\angle{O_1AO_2}
∠
O
1
A
O
2
is an obtuse angle. Let the circumcircle of
Δ
O
1
A
O
2
\Delta{O_1AO_2}
Δ
O
1
A
O
2
intersect
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
respectively in points
C
(
≠
A
)
C (\neq A)
C
(
=
A
)
and
D
(
≠
A
)
D (\neq A)
D
(
=
A
)
. Let the line
C
B
CB
CB
intersect
Γ
2
\Gamma_2
Γ
2
in
E
E
E
; let the line
D
B
DB
D
B
intersect
Γ
1
\Gamma_1
Γ
1
in
F
F
F
. Prove that, the points
C
,
D
,
E
,
F
C, D, E, F
C
,
D
,
E
,
F
are concyclic.
geometry