Curves A, B, C and D are defined in the plane as follows:
(1987 William Lowell Putnam Mathematics Competition)
Let . The equations defining A and B are the real and imaginary parts of the equation , and similarly the equations defining C and D are the real and imaginary parts of . Hence, for all real x and y, we have if and only if . This is equivalent to , that is, .
Isn’t that an elegant solution? What do you think?
PS: Solution published in “Complex Numbers from A to …Z” by Titu Andreescu and Dorin Andrica