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Sharygin Geometry Olympiad
2019 Sharygin Geometry Olympiad
23
23
Part of
2019 Sharygin Geometry Olympiad
Problems
(1)
Sharygin CR 2019 P23
Source: Sharygin CR 2019 P23 (Grade 10 - 11)
3/6/2019
In the plane, let
a
a
a
,
b
b
b
be two closed broken lines (possibly self-intersecting), and
K
K
K
,
L
L
L
,
M
M
M
,
N
N
N
be four points. The vertices of
a
a
a
,
b
b
b
and the points
K
K
K
L
L
L
,
M
M
M
,
N
N
N
are in general position (i.e. no three of these points are collinear, and no three segments between them concur at an interior point). Each of segments
K
L
KL
K
L
and
M
N
MN
MN
meets
a
a
a
at an even number of points, and each of segments
L
M
LM
L
M
and
N
K
NK
N
K
meets
a
a
a
at an odd number of points. Conversely, each of segments
K
L
KL
K
L
and
M
N
MN
MN
meets
b
b
b
at an odd number of points, and each of segments
L
M
LM
L
M
and
N
K
NK
N
K
meets
b
b
b
at an even number of points. Prove that
a
a
a
and
b
b
b
intersect.
geometry