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HMMT 二月 2007 · 冲刺赛 · 第 3 题

HMMT February 2007 — Guts Round — Problem 3

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

  1. [ 5 ] The equation x + 2 x = i has two complex solutions. Determine the product of their real parts.
    10 HARVARD-MIT MATHEMATICS TOURNAMENT, 24 FEBRUARY 2007 — GUTS ROUNDth

英文原题

[ 5 ] The equation x 2 + 2 x = i has two complex solutions. Determine the product of their real parts.
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10 th HARVARD-MIT MATHEMATICS TOURNAMENT, 24 FEBRUARY 2007 — GUTS ROUND

解析

英文解析

  1. [ 5 ] The equation x + 2 x = i has two complex solutions. Determine the product of their real parts.
    √
    √
    1 − 2 2 iπ
    Answer: . Complete the square by adding 1 to each side. Then ( x + 1) = 1 + i = e 2, so 4
    √2
    4 iπx + 1 = ± e 2. The desired product is then 8
    ( ( ))
    √
    ( ( ) ) ( ( ) ) ( )π
    √ √ √ √
    1 + cosπ π π 1 − 2
    4 4
    2 4 − 1 + cos 2 − 1 − cos 2 = 1 − cos 2 = 1 − 2 = .
    8 8 8 2 2
    10 HARVARD-MIT MATHEMATICS TOURNAMENT, 24 FEBRUARY 2007 — GUTS ROUNDth