MathDB

To dissect a polygon means to divide it into several regions by cutting along finitely many line segments. For example, the diagram below shows a dissection of a hexagon into two triangles and two quadrilaterals: https://cdn.artofproblemsolving.com/attachments/0/a/378e477bcbcec26fc90412c3eada855ae52b45.png An integer-ratio right triangle is a right triangle whose side lengths are in an integer ratio. For example, a triangle with sides

3,4,5

is an integer-ratio right triangle, and so is a triangle with sides

\frac52 \sqrt3 ,6\sqrt3, \frac{13}{2} \sqrt3

. On the other hand, the right triangle with sides

\sqrt2 ,\sqrt5, \sqrt7

is not an integer-ratio right triangle. Determine, with proof, all integers

n

for which it is possible to completely dissect a regular

n

-sided polygon into integer-ratio right triangles.

Problem Statement