PUMaC 2017 · 代数（A 组） · 第 2 题
PUMaC 2017 — Algebra (Division A) — Problem 2
题目详情
- Suppose z = 2 + 2 i , where i = − 1. The product of all possible values of the real part of z p can be written in the form where p and q are relatively prime positive integers. Find p + q . q
解析
- Let w = z + z . Then, we have 3 3 3 3 2 w = ( z + z ) = z + 3 zz ( z + z ) + z = (2 + 2 i ) + 3 | z | w + (2 − 2 i ) = 4 + 6 w, w 1 so the product of all possible values of w is 4. Since the real part of z is , the answer is , 2 2 which gives 3 . Problem written by Matt Tyler