Problem:
Curves A, B, C and D are defined in the plane as follows:
–
.
Prove that
(1987 William Lowell Putnam Mathematics Competition)
Solution:
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,
.
Thus, .
Isn’t that an elegant solution? What do you think?
Nalin Pithwa
PS: Solution published in “Complex Numbers from A to …Z” by Titu Andreescu and Dorin Andrica