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Kazakhstan National Olympiad
2018 Kazakhstan National Olympiad
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1
Part of
2018 Kazakhstan National Olympiad
Problems
(1)
Kazakhstan MO 2018
Source: Kazakhstan MO 2018 final round.Grade 11;Problem 1
5/4/2018
In an equilateral trapezoid, the point
O
O
O
is the midpoint of the base
A
D
AD
A
D
. A circle with a center at a point
O
O
O
and a radius
B
O
BO
BO
is tangent to a straight line
A
B
AB
A
B
. Let the segment
A
C
AC
A
C
intersect this circle at point
K
(
K
≠
C
)
K(K \ne C)
K
(
K
=
C
)
, and let
M
M
M
is a point such that
A
B
C
M
ABCM
A
BCM
is a parallelogram. The circumscribed circle of a triangle
C
M
D
CMD
CM
D
intersects the segment
A
C
AC
A
C
at a point
L
(
L
≠
C
)
L(L\ne C)
L
(
L
=
C
)
. Prove that
A
K
=
C
L
AK=CL
A
K
=
C
L
.
geometry
Kazakhstan