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Problems
Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Kyiv City MO
2021 Kyiv City MO
2021 Kyiv City MO Round 1
10.3
10.3
Part of
2021 Kyiv City MO Round 1
Problems
(1)
Circles and parallelogram
Source: Kyiv City MO 2021 Round 1, Problem 10.3
12/21/2023
Circles
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
with centers at points
O
1
O_1
O
1
and
O
2
O_2
O
2
intersect at points
A
A
A
and
B
B
B
. Let point
C
C
C
be such that
A
O
2
C
O
1
AO_2CO_1
A
O
2
C
O
1
is a parallelogram. An arbitrary line is drawn through point
A
A
A
, which intersects the circles
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
at points
X
X
X
and
Y
Y
Y
, respectively. Prove that
C
X
=
C
Y
CX = CY
CX
=
C
Y
.Proposed by Oleksii Masalitin
geometry
parallelogram