MathDB
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Contests
National and Regional Contests
Azerbaijan Contests
JBMO TST - Azerbaijan
2015 Azerbaijan JBMO TST
2015 Azerbaijan JBMO TST
Part of
JBMO TST - Azerbaijan
Subcontests
(1)
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AZE JBMO TST
Let
A
B
C
ABC
A
BC
be a triangle such that
A
B
AB
A
B
is not equal to
A
C
AC
A
C
. Let
M
M
M
be the midpoint of
B
C
BC
BC
and
H
H
H
be the orthocenter of triangle
A
B
C
ABC
A
BC
. Let
D
D
D
be the midpoint of
A
H
AH
A
H
and
O
O
O
the circumcentre of triangle
B
C
H
BCH
BC
H
. Prove that
D
A
M
O
DAMO
D
A
MO
is a parallelogram.
AZE JBMO TST
Acute-angled
△
A
B
C
\triangle{ABC}
△
A
BC
triangle with condition
A
B
<
A
C
<
B
C
AB<AC<BC
A
B
<
A
C
<
BC
has cimcumcircle
C
,
C^,
C
,
with center
O
O
O
and radius
R
R
R
.And
B
D
BD
B
D
and
C
E
CE
CE
diametrs drawn.Circle with center
O
O
O
and radius
R
R
R
intersects
A
C
AC
A
C
at
K
K
K
.And circle with center
A
A
A
and radius
A
D
AD
A
D
intersects
B
A
BA
B
A
at
L
L
L
.Prove that
E
K
EK
E
K
and
D
L
DL
D
L
lines intersects at circle
C
,
C^,
C
,
.
AZE JBMO TST
There is a triangle
A
B
C
ABC
A
BC
that
A
B
AB
A
B
is not equal to
A
C
AC
A
C
.
B
D
BD
B
D
is interior bisector of
∠
A
B
C
\angle{ABC}
∠
A
BC
(
D
∈
A
C
D\in AC
D
∈
A
C
)
M
M
M
is midpoint of
C
B
A
CBA
CB
A
arc.Circumcircle of
△
B
D
M
\triangle{BDM}
△
B
D
M
cuts
A
B
AB
A
B
at
K
K
K
and
J
,
J,
J
,
is symmetry of
A
A
A
according
K
K
K
.If
D
J
∩
A
M
=
(
O
)
DJ\cap AM=(O)
D
J
∩
A
M
=
(
O
)
, Prove that
J
,
B
,
M
,
O
J,B,M,O
J
,
B
,
M
,
O
are cyclic.