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Iranian Geometry Olympiad
2015 Iran Geometry Olympiad
1
IGO - Hard level - problem 1
IGO - Hard level - problem 1
Source:
September 3, 2015
IGO
geometry
Problem Statement
let
w
1
w_1
w
1
and
w
2
w_2
w
2
two circles such that
w
1
∩
w
2
=
{
A
,
B
}
w_1 \cap w_2 = \{ A , B \}
w
1
∩
w
2
=
{
A
,
B
}
let
X
X
X
a point on
w
2
w_2
w
2
and
Y
Y
Y
on
w
1
w_1
w
1
such that
B
Y
⊥
B
X
BY \bot BX
B
Y
⊥
BX
suppose that
O
O
O
is the center of
w
1
w_1
w
1
and
X
′
=
w
2
∩
O
X
X' = w_2 \cap OX
X
′
=
w
2
∩
OX
now if
K
=
w
2
∩
X
′
Y
K = w_2 \cap X'Y
K
=
w
2
∩
X
′
Y
prove
X
X
X
is the midpoint of arc
A
K
AK
A
K
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