NZMO Round One

Geometry Difficulty 4.8 AIME Prove it New Zealand

Problem:
You have an unlimited supply of square tiles with side length $1$ and equilateral triangle tiles with side length $1$ . For which $n$ can you use these tiles to create a convex $n$ -sided polygon? The tiles must fit together without gaps and may not overlap.

Solution

Solution:
All the angles in squares and equilateral triangles are multiples of $30^{\circ}$ . So all the external angles of the $n$ -sided polygon are multiples of $30^{\circ}$ . Since the polygon is convex, this implies that all external angles are greater than or equal to $30^{\circ}$ . However, the sum of the external angles is $360^{\circ}$ , therefore
$n \times 30^{\circ} \leq 360^{\circ}.$
Hence $n \leq 12$ . Also all polygons have at least $3$ sides so $3 \leq n \leq 12$ . Finally we demonstrate that it is possible for any $3 \leq n \leq 12$ using the following illustrations.

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

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