25 lines
750 B
Typst
25 lines
750 B
Typst
#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.
|
|
])
|