HMMT 二月 2009 · 冲刺赛 · 第 23 题
HMMT February 2009 — Guts Round — Problem 23
题目详情
英文原题
- [ 10 ] The roots of z + z + z + 1 = 0 are the vertices of a convex polygon in the complex plane. Findthe sum of the squares of the side lengths of the polygon.
解析
英文解析
- [ 10 ] The roots of z + z + z + 1 = 0 are the vertices of a convex polygon in the complex plane. Findthe sum of the squares of the side lengths of the polygon.
√
Answer: 12 − 4 2
4 2 ± iπ/ 4
Solution: Factoring the polynomial as ( z + 1)( z + 1) = 0, we find that the 6 roots are e ,
± iπ/ 2 ± i 3 π/ 4
e , e . The calculation then follows from the Law of Cosines or the distance formula.