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National and Regional Contests
India Contests
India Pre-Regional Mathematical Olympiad
2013 India PRMO
17
17
Part of
2013 India PRMO
Problems
(1)
2013 preRMO p17, 3 circles, tangent, radius wanted
Source:
8/8/2019
Let
S
S
S
be a circle with centre
O
O
O
. A chord
A
B
AB
A
B
, not a diameter, divides
S
S
S
into two regions
R
1
R_1
R
1
and
R
2
R_2
R
2
such that
O
O
O
belongs to
R
2
R_2
R
2
. Let
S
1
S_1
S
1
be a circle with centre in
R
1
R_1
R
1
, touching
A
B
AB
A
B
at
X
X
X
and
S
S
S
internally. Let
S
2
S_2
S
2
be a circle with centre in
R
2
R_2
R
2
, touching
A
B
AB
A
B
at
Y
Y
Y
, the circle
S
S
S
internally and passing through the centre of
S
S
S
. The point
X
X
X
lies on the diameter passing through the centre of
S
2
S_2
S
2
and
∠
Y
X
O
=
3
0
o
\angle YXO=30^o
∠
Y
XO
=
3
0
o
. If the radius of
S
2
S_2
S
2
is
100
100
100
then what is the radius of
S
1
S_1
S
1
?
geometry
circles
radius