HMMT 二月 2016 · 代数 · 第 1 题

HMMT February 2016 — Algebra — Problem 1

Let z be a complex number such that | z | = 1 and | z − 1 . 45 | = 1 . 05. Compute the real part of z . n

解析

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 20 Answer: 29 From the problem, let A denote the point z on the unit circle, B denote the point 1 . 45 on the real axis, and O the origin. Let AH be the height of the triangle OAH and H lies on the segment OB . The real part 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