Problem:
How many non-zero complex numbers have the property that , and , when represented by points in the complex plane, are the three distinct vertices of an equilateral triangle?
Answer Choices:
A.
B.
C.
D.
E. infinitely many
Solution:
Convert and into modulus-argument (polar) form, giving for some and . Thus, by De Moivre's Theorem, . Since the distance from to is , and the triangle is equilateral, the distance from to must also be , so , giving . (We know since the problem statement specifies that must be nonzero.)
Now, to get from to , which should be a rotation of if the triangle is equilateral, we multiply by , again using De Moivre's Theorem. Thus we require (where can be any integer). If , we must have , while if , we must have . Hence there are values that work for . By symmetry, the interval will also give solutions. The answer is thus .
OR
As before, . Represent in polar form. By De Moivre's Theorem, . To form an equilateral triangle, their difference in angle must be , so
From the polar form of , we know that , so cis cycles in a circle twice. By contrast, represent fixed, distinct points. Thus, cis(2 ) intersects these points twice each
The problems on this page are the property of the MAA's American Mathematics Competitions