HMMT 二月 2012 · 代数 · 第 4 题

英文解析

4. During the weekends, Eli delivers milk in the complex plane. On Saturday, he begins at z and delivers
   3 5 7 2013
   milk to houses located at z , z , z , . . . , z , in that order; on Sunday, he begins at 1 and delivers
   2 4 6 2012
   milk to houses located at z , z , z , . . . , z , in that order. Eli always walks directly (in a straightline) between two houses. If the distance he must travel from his starting point to the last house is
   √
   2012 on both days, find the real part of z .2
   1005
   Answer: Note that the distance between two points in the complex plane, m and n , is | m − n | .
   1006
   We have that
   1006 1006
   ∑ ∑
   ∣ ∣ ∣ ∣ √
   2 k +1 2 k − 1 2 k 2 k − 2
   ∣ ∣ ∣ ∣
   z − z = z − z = 2012 .
   k =1 k =1
   Algebra Test
   However, noting that
   1006 1006
   ∑ ∑
   ∣ ∣ ∣ ∣
   2 k 2 k − 2 2 k +1 2 k − 1
   ∣ ∣ ∣ ∣
   | z | · z − z = z − z ,
   k =1 k =1
   √
   we must have | z | = 1. Then, since Eli travels a distance of 2012 on each day, we have
   1006 1006 1006
   ∑ ∑ ∑
   ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
   2 k − 2
   2 k 2 k − 2 2 2 k − 2 2
   ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
   z − z = z − 1 · z = z − 1 · | z |
   k =1 k =1 k =1
   √
   ∣ ∣
   ∣ ∣2 = 1006 z − 1 = 2012 ,
   √
   ∣ ∣
   2012
   2 2
   ∣ ∣
   so z − 1 = . Since | z | = 1, we can write z = cos ( θ ) + i sin ( θ ) and then z = cos (2 θ ) + i sin (2 θ ).
   1006
   Hence,
   √
   √
   √
   ∣ ∣
   2012
   2 22
   ∣ ∣
   z − 1 = (cos (2 θ ) − 1) + sin (2 θ ) = 2 − 2 cos (2 θ ) = ,
   1006
   2 2 1005
   so 2 − 2 cos (2 θ ) = . The real part of z , cos (2 θ ), is thus .
   1006 1006