Warm-Up: Pairs

src/Warm-Ups/Pairs/main.typ

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+#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.
+])

src/Warm-Ups/Pairs/meta.toml

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+[metadata]
+title = "Pairs"
+
+[publish]
+handout = true
+solutions = true