Problem:
Let w=2+i. What real number r has the property that r, w, and w2 are three collinear points in the complex plane?
Answer Choices:
A. 43
B. 1
C. 57
D. 23
E. 35
Solution:
We need 2+i,(2+i)2=3+4i and r to be collinear in the complex plane, which is equivalent to ensuring that the points (2,1),(3,4),(r,0) are collinear.
Equating on slopes:
3−24−1=r−30−4→3(r−3)=−4→r=35→(E).
The problems on this page are the property of the MAA's American Mathematics Competitions