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National and Regional Contests
Iran Contests
Iran MO (2nd Round)
2020 Iran MO (2nd Round)
P3
P3
Part of
2020 Iran MO (2nd Round)
Problems
(1)
beautiful geometry problem.
Source: Iranian second round 2020 day1 P3
7/14/2020
let
ω
1
\omega_1
ω
1
be a circle with
O
1
O_1
O
1
as its center , let
ω
2
\omega_2
ω
2
be a circle passing through
O
1
O_1
O
1
with center
O
2
O_2
O
2
let
A
A
A
be one of the intersection of
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
let
x
x
x
be a line tangent line to
ω
1
\omega_1
ω
1
passing from
A
A
A
let
ω
3
\omega_3
ω
3
be a circle passing through
O
1
,
O
2
O_1,O_2
O
1
,
O
2
with its center on the line
x
x
x
and intersect
ω
2
\omega_2
ω
2
at
P
P
P
(not
O
1
O_1
O
1
) prove that the reflection of
P
P
P
through
x
x
x
is on
ω
1
\omega_1
ω
1
geometry