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National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2016 Moldova Team Selection Test
7
7
Part of
2016 Moldova Team Selection Test
Problems
(1)
geo... again >:(
Source: MDA TST 2016 P7
3/26/2019
Let
Ω
\Omega
Ω
and
O
O
O
be the circumcircle of acute triangle
A
B
C
ABC
A
BC
and its center, respectively.
M
≠
O
M\ne O
M
=
O
is an arbitrary point in the interior of
A
B
C
ABC
A
BC
such that
A
M
AM
A
M
,
B
M
BM
BM
, and
C
M
CM
CM
intersect
Ω
\Omega
Ω
at
A
1
A_{1}
A
1
,
B
1
B_{1}
B
1
, and
C
1
C_{1}
C
1
, respectiuvely. Let
A
2
A_{2}
A
2
,
B
2
B_{2}
B
2
, and
C
2
C_{2}
C
2
be the circumcenters of
M
B
C
MBC
MBC
,
M
C
A
MCA
MC
A
, and
M
A
B
MAB
M
A
B
, respectively. It is to be proven that
A
1
A
2
A_{1}A_{2}
A
1
A
2
,
B
1
B
2
B_{1}B_{2}
B
1
B
2
,
C
1
C
2
C_{1}C{2}
C
1
C
2
concur.
geometry
circumcircle