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National and Regional Contests
Russia Contests
All-Russian Olympiad Regional Round
1995 All-Russian Olympiad Regional Round
11.7
11.7
Part of
1995 All-Russian Olympiad Regional Round
Problems
(1)
MN = AE +AF, 2 intersecting circles
Source: All-Russian MO 1995 Regional (R4) 11.7
8/26/2024
Circles
S
1
S_1
S
1
and
S
2
S_2
S
2
with centers
O
1
O_1
O
1
and
O
2
O_2
O
2
respectively intersect at
A
A
A
and
B
B
B
. Ray
O
1
B
O_1B
O
1
B
meets
S
2
S_2
S
2
again at
F
F
F
, and ray
O
2
B
O_2B
O
2
B
meets
S
1
S_1
S
1
again at
E
E
E
. The line through
B
B
B
parallel to
E
F
EF
EF
intersects
S
1
S_1
S
1
and
S
2
S_2
S
2
again at
M
M
M
and
N
N
N
, respectively. Prove that
M
N
=
A
E
+
A
F
MN = AE +AF
MN
=
A
E
+
A
F
.
geometry
equal segments