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National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
2017 Bulgaria National Olympiad
1
1
Part of
2017 Bulgaria National Olympiad
Problems
(1)
A cyclic quadrilateral
Source: Bulgarian NMO 2017, 3rd round, p.1
4/22/2017
An convex qudrilateral
A
B
C
D
ABCD
A
BC
D
is given.
O
O
O
is the intersection point of the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
. The points
A
1
,
B
1
,
C
1
,
D
1
A_1,B_1,C_1, D_1
A
1
,
B
1
,
C
1
,
D
1
lie respectively on
A
O
,
B
O
,
C
O
,
D
O
AO, BO, CO, DO
A
O
,
BO
,
CO
,
D
O
such that
A
A
1
=
C
C
1
,
B
B
1
=
D
D
1
AA_1=CC_1, BB_1=DD_1
A
A
1
=
C
C
1
,
B
B
1
=
D
D
1
. The circumcircles of
△
A
O
B
\triangle AOB
△
A
OB
and
△
C
O
D
\triangle COD
△
CO
D
meet at second time at
M
M
M
and the the circumcircles of
△
A
O
D
\triangle AOD
△
A
O
D
and
△
B
O
C
\triangle BOC
△
BOC
- at
N
N
N
. The circumcircles of
△
A
1
O
B
1
\triangle A_1OB_1
△
A
1
O
B
1
and
△
C
1
O
D
1
\triangle C_1OD_1
△
C
1
O
D
1
meet at second time at
P
P
P
and the the circumcircles of
△
A
1
O
D
1
\triangle A_1OD_1
△
A
1
O
D
1
and
△
B
1
O
C
1
\triangle B_1OC_1
△
B
1
O
C
1
- at
Q
Q
Q
. Prove that the quadrilateral
M
N
P
Q
MNPQ
MNPQ
is cyclic.
geometry