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

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