HMMT 二月 2016 · 代数 · 第 1 题

英文解析

1. Let z be a complex number such that | z | = 1 and | z − 1 . 45 | = 1 . 05. Compute the real part of z .
   Proposed by: Evan Chen
   Answer:20
   From the problem, let A denote the point z on the unit circle, B denote the point 1 . 45 on the real axis,29
   and O the origin. Let AH be the height of the triangle OAH and H lies on the segment OB . The realpart of z is OH . Now we have OA = 1 , OB = 1 . 45, and AB = 1 . 05. Thus
   2 2 2
   1 + 1 . 45 − 1 . 05 20
   OH = OA cos ∠ AOB = cos ∠ AOB = = .
   2 · 1 · 1 . 45 29 n