jmc

Question

During the weekends, Eli delivers milk in the complex plane. On Saturday, he begins at z and delivers milk to houses located at z^3,z^5,z^7,,z^2013 in that order; on Sunday, he begins at 1 and delivers milk to houses located at z^2,z^4,z^6,,z^2012 in that order. Eli always walks directly (in a straight line) 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.

Accepted Answer

The sum of the Saturday distances is |z^3 - z| + |z^5 - z^3| + + |z^2013 - z^2011| = 2012.The sum of the Sunday distances is |z^2 - 1| + |z^4 - z^2| + + |z^2012 - z^2010| = 2012.Note that |z^3 - z| + |z^5 - z^3| + + |z^2013 - z^2011| = |z| (|z^2 - 1| + |z^4 - z^2| + + |z^2012 - z^2010|),so |z| = 1. Then align* |z^2 - 1| + |z^4 - z^2| + + |z^2012 - z^2010| &= |z^2 - 1| + |z^2| |z^2 - 1| + + |z^2010| |z^2 - 1| &= |z^2 - 1| + |z|^2 |z^2 - 1| + + |z|^2010 |z^2 - 1| &= 1006 |z^2 - 1|, align*so |z^2 - 1| = 20121006.We have that |z^2| = |z|^2 = 1. Let z^2 = a + bi, where a and b are real numbers, so a^2 + b^2 = 1. From the equation |z^2 - 1| = 20121006, (a - 1)^2 + b^2 = 20121006^2 = 1503.Subtracting these equations, we get 2a - 1 = 1 - 1503 = 502503,so a = 10051006.