diff --git a/src/Warm-Ups/Pairs/main.typ b/src/Warm-Ups/Pairs/main.typ new file mode 100644 index 0000000..847c4ed --- /dev/null +++ b/src/Warm-Ups/Pairs/main.typ @@ -0,0 +1,24 @@ +#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. +]) diff --git a/src/Warm-Ups/Pairs/meta.toml b/src/Warm-Ups/Pairs/meta.toml new file mode 100644 index 0000000..fa8d226 --- /dev/null +++ b/src/Warm-Ups/Pairs/meta.toml @@ -0,0 +1,6 @@ +[metadata] +title = "Pairs" + +[publish] +handout = true +solutions = true