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Problems
Contests
International Contests
Gulf Math Olympiad
2017 Gulf Math Olympiad
3
3
Part of
2017 Gulf Math Olympiad
Problems
(1)
so much circles
Source:
9/28/2017
Let
C
1
C_1
C
1
and
C
2
C_2
C
2
be two different circles , and let their radii be
r
1
r_1
r
1
and
r
2
r_2
r
2
, the two circles are passing through the two points
A
A
A
and
B
B
B
(i)Let
P
1
P_1
P
1
on
C
1
C_1
C
1
and
P
2
P_2
P
2
on
C
2
C_2
C
2
such that the line
P
1
P
2
P_1P_2
P
1
P
2
passes through
A
A
A
. Prove that
P
1
B
⋅
r
2
=
P
2
B
⋅
r
1
P_1B \cdot r_2 = P_2B \cdot r_1
P
1
B
⋅
r
2
=
P
2
B
⋅
r
1
(ii)Let
D
E
F
DEF
D
EF
be a triangle that it's inscribed in
C
1
C_1
C
1
, and let
D
′
E
′
F
′
D'E'F'
D
′
E
′
F
′
be a triangle that is inscribed in
C
2
C_2
C
2
. The lines
E
E
′
EE'
E
E
′
,
D
D
′
DD'
D
D
′
and
F
F
′
FF'
F
F
′
all pass through
A
A
A
. Prove that the triangles
D
E
F
DEF
D
EF
and
D
′
E
′
F
′
D'E'F'
D
′
E
′
F
′
are similar(iii)The circle
C
3
C_3
C
3
also passes through
A
A
A
and
B
B
B
. Let
l
l
l
be a line that passes through
A
A
A
and cuts circles
C
i
C_i
C
i
in
M
i
M_i
M
i
with
i
=
1
,
2
,
3
i = 1,2,3
i
=
1
,
2
,
3
. Prove that the value of
M
1
M
2
M
1
M
3
\frac{M_1M_2}{M_1M_3}
M
1
M
3
M
1
M
2
is constant regardless of the position of
l
l
l
Provided that
l
l
l
is different from
A
B
AB
A
B
geometry
circles