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National and Regional Contests
Paraguay Contests
Paraguay Mathematical Olympiad
2006 Paraguay Mathematical Olympiad
3
3
Part of
2006 Paraguay Mathematical Olympiad
Problems
(1)
Problem 3
Source: Paraguayan Mathematical Olympiad 2006
7/22/2015
Let
Γ
A
\Gamma_A
Γ
A
,
Γ
B
\Gamma_B
Γ
B
,
Γ
C
\Gamma_C
Γ
C
be circles such that
Γ
A
\Gamma_A
Γ
A
is tangent to
Γ
B
\Gamma_B
Γ
B
and
Γ
B
\Gamma_B
Γ
B
is tangent to
Γ
C
\Gamma_C
Γ
C
. All three circles are tangent to lines
L
L
L
and
M
M
M
, with
A
A
A
,
B
B
B
,
C
C
C
being the tangency points of
M
M
M
with
Γ
A
\Gamma_A
Γ
A
,
Γ
B
\Gamma_B
Γ
B
,
Γ
C
\Gamma_C
Γ
C
, respectively. Given that
12
=
r
A
<
r
B
<
r
C
=
75
12=r_A<r_B<r_C=75
12
=
r
A
<
r
B
<
r
C
=
75
, calculate:a) the length of
r
B
r_B
r
B
. b) the distance between point
A
A
A
and the point of intersection of lines
L
L
L
and
M
M
M
.
geometry