Problem:
Let be a complex number with . Let be the polygon in the complex plane whose vertices are and every such that . Then the area enclosed by can be written in the form , where is an integer. Find the remainder when is divided by .
Solution:
Suppose that satisfies the equation. Then . Multiply both sides of the equation by to get . Thus, is times a cube root of . This means that is an equilateral triangle inscribed in a circle of radius . The area of such a triangle is . So the requested remainder is .
The problems on this page are the property of the MAA's American Mathematics Competitions