#import "@local/handout:0.1.0": *
#import "@preview/cetz:0.4.2"

#show: handout.with(
  title: [Warm-Up: Pairs],
  by: "Mark",
)

#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 line segments 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.
])