Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 6.0 National Olympiad Prove it United States

Problem:

Consider a sheet of paper in the shape of an equilateral triangle creased along the dashed lines as in the figure below on the left. Folding over each of the three corners along the dashed lines creates a new object which is uniformly four layers thick, as in the figure below on the right. The number in each region indicates that region's thickness (in layers of paper).

We have just seen one example of how a plane figure can be folded into an object with a uniform thickness. This problem asks you to produce several other examples. In each case, you may fold along any lines. The different parts that are folded may or may not be congruent. Assume that paper may be folded any number of times without tearing or becoming too thick to fold. If needed, you can use any of the following tools:
- a magic ruler with which you can draw a line through any two given points and you can split any segment into as many equal parts as you wish; and
- a right triangle tool with which you can drop perpendiculars from points to lines and erect perpendiculars to lines from points on them.

Given these rules:

a) Show how to fold an equilateral triangle into an object with a uniform thickness of 3 layers.

b) Show how to fold a $30^{\circ}$ - $60^{\circ}$ - $90^{\circ}$ triangle into an object with a uniform thickness of 3 layers.

c) Show that every triangle can be folded into an object with a uniform thickness of 2020 layers.

Solution

Solution:

a. Divide the sides of the triangle into three equal parts and connect them with lines as in the figure on the left below.

Now crease and fold along the dotted lines to form the figure in the center. Finally, fold along the dashed lines in the center figure to form the figure on the right.

b. Divide the triangle into three triangles as shown in the figure on the left below. (The upper dashed line is the perpendicular bisector of the hypotenuse and its intersection with the vertical edge is connected to the lower-right vertex of the triangle.)

Now crease and fold along the two dashed lines to achieve the figure on the right.

c. In fact, any triangle can be folded to a uniform thickness of $2n$ layers for any positive integer $n$ . Label the vertices of the triangle $A, B, C$ so that $AB$ is the longest side, guaranteeing that $\angle A$ and $\angle B$ are acute. Let points $D$ and $E$ bisect the segments $AC$ and $BC$ . Next drop perpendiculars from $D$ and $E$ to $AB$ and fold along the three dashed lines to form the rectangle in the upper right of the figure below.

That rectangle is 2 layers thick. To make an object of $2n$ layers, divide the rectangle's top and bottom edges into $n$ equal pieces and connect pairs of points to make $n$ regions. Folding along all $n-1$ dashed lines will produce an object with $2n$ layers. The figure demonstrates how to do this if $n=3$ , yielding an object with $2n=6$ layers. For 2020 layers, let $n=1010$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.