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Contests
National and Regional Contests
Bosnia Herzegovina Contests
JBMO TST - Bosnia and Herzegovina
2018 Bosnia and Herzegovina Junior BMO TST
3
3
Part of
2018 Bosnia and Herzegovina Junior BMO TST
Problems
(1)
P3 Bosnia and Herzegovina JBMO TST
Source: Bosnia and Herzegovina JBMO TST 2018
7/14/2018
Let
Γ
\Gamma
Γ
be circumscribed circle of triangle
A
B
C
ABC
A
BC
(
A
B
≠
A
C
)
(AB \neq AC)
(
A
B
=
A
C
)
. Let
O
O
O
be circumcenter of the triangle
A
B
C
ABC
A
BC
. Let
M
M
M
be a point where angle bisector of angle
B
A
C
BAC
B
A
C
intersects
Γ
\Gamma
Γ
. Let
D
D
D
(
D
≠
M
)
(D \neq M)
(
D
=
M
)
be a point where circumscribed circle of the triangle
B
O
M
BOM
BOM
intersects line segment
A
M
AM
A
M
and let
E
E
E
(
E
≠
M
)
(E \neq M)
(
E
=
M
)
be a point where circumscribed circle of triangle
C
O
M
COM
COM
intersects line segment
A
M
AM
A
M
. Prove that
B
D
+
C
E
=
A
M
BD+CE=AM
B
D
+
CE
=
A
M
.
geometry