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Problems
Contests
National and Regional Contests
Canada Contests
Canadian Open Math Challenge
2024 Canadian Open Math Challenge
C3
C3
Part of
2024 Canadian Open Math Challenge
Problems
(1)
2024 COMC C3
Source:
11/4/2024
Let
A
B
C
ABC
A
BC
be a triangle for which the tangent from
A
A
A
to the circumcircle intersects line
B
C
BC
BC
at
D
D
D
, and let
O
O
O
be the circumcenter. Construct the line
l
l
l
that passes through
A
A
A
and is perpendicular to
O
D
OD
O
D
.
l
l
l
intersects
O
D
OD
O
D
at
E
E
E
and
B
C
BC
BC
at
F
F
F
. Let the circle passing through
A
D
O
ADO
A
D
O
intersect
B
C
BC
BC
again at
H
H
H
. It is given that
A
D
=
A
O
=
1
AD=AO=1
A
D
=
A
O
=
1
.a) Find
O
E
OE
OE
b) Suppose for this part only that
F
H
=
1
12
FH=\frac{1}{\sqrt{12}}
F
H
=
12
1
: determine the area of triangle
O
E
F
OEF
OEF
. c) Suppose for this part only that
B
C
=
3
BC=\sqrt3
BC
=
3
: determine the area of triangle
O
E
F
OEF
OEF
. d) Suppose that
B
B
B
lies on the angle bisector of
D
E
F
DEF
D
EF
. Find the area of the triangle
O
E
F
OEF
OEF
.
Comc