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National and Regional Contests
Iran Contests
Iran MO (2nd Round)
2023 Iran MO (2nd Round)
P6
P6
Part of
2023 Iran MO (2nd Round)
Problems
(1)
2023 Iran MO 2nd round P6
Source: 2023 Iran MO
5/17/2023
6. Circles
W
1
W_{1}
W
1
and
W
2
W_{2}
W
2
with equal radii are given. Let
P
P
P
,
Q
Q
Q
be the intersection of the circles. points
B
B
B
and
C
C
C
are on
W
1
W_{1}
W
1
and
W
2
W_{2}
W
2
such that they are inside
W
2
W_{2}
W
2
and
W
1
W_{1}
W
1
respectively. Points
X
X
X
,
Y
Y
Y
≠
\neq
=
P
P
P
are on
W
1
W_{1}
W
1
and
W
2
W_{2}
W
2
respectively, such that
∠
B
P
Q
=
∠
B
Y
Q
\angle{BPQ}=\angle{BYQ}
∠
BPQ
=
∠
B
Y
Q
and
∠
C
P
Q
=
∠
C
X
Q
\angle{CPQ}=\angle{CXQ}
∠
CPQ
=
∠
CXQ
.Denote by
S
S
S
as the other intersection of
(
Y
P
B
)
(YPB)
(
Y
PB
)
and
(
X
P
C
)
(XPC)
(
XPC
)
. Prove that
Q
S
,
B
C
,
X
Y
QS,BC,XY
QS
,
BC
,
X
Y
are concurrent.
geometry