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#problem()
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#problem()
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$n$ black and $n$ white points are randomly distributed on a plane. No three points are collinear.\
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$n$ black and $n$ white points are randomly distributed on a plane. No three points are collinear.\
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Show that it is always possible to draw $n$ nonintersecting lines between pairs of points of different colors.
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Show that it is always possible draw $n$ nonintersecting lines between pairs of points of different colors.
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#solution([
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Consider the total length of all lines on the plane.
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#v(2mm)
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If we replace a pair of intersecting lines with two nonintersecting lines, \
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we strictly decrease this total length (by the triangle inequality).
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#v(2mm)
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Thus, the arrangement of lines with the minimum total length must not have any intersections. \
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Showing that a minimum exists is fairly easy.
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])
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