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Problems
Contests
National and Regional Contests
India Contests
Regional Mathematical Olympiad
2017 India Regional Mathematical Olympiad
5
5
Part of
2017 India Regional Mathematical Olympiad
Problems
(1)
RMO 2017 P5
Source: RMO 2017 P5
10/8/2017
Let
Ω
\Omega
Ω
be a circle with a chord
A
B
AB
A
B
which is not a diameter.
Γ
1
\Gamma_{1}
Γ
1
be a circle on one side of
A
B
AB
A
B
such that it is tangent to
A
B
AB
A
B
at
C
C
C
and internally tangent to
Ω
\Omega
Ω
at
D
D
D
. Likewise, let
Γ
2
\Gamma_{2}
Γ
2
be a circle on the other side of
A
B
AB
A
B
such that it is tangent to
A
B
AB
A
B
at
E
E
E
and internally tangent to
Ω
\Omega
Ω
at
F
F
F
. Suppose the line
D
C
DC
D
C
intersects
Ω
\Omega
Ω
at
X
≠
D
X \neq D
X
=
D
and the line
F
E
FE
FE
intersects
Ω
\Omega
Ω
at
Y
≠
F
Y \neq F
Y
=
F
. Prove that
X
Y
XY
X
Y
is a diameter of
Ω
\Omega
Ω
.
geometry
circles