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1d8782a03b Warm-Up: Georgian
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2025-10-26 11:12:42 -07:00
48e7abd47e Warm-Up: Pairs 2025-10-26 11:12:39 -07:00

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