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HMMT 二月 2008 · 冲刺赛 · 第 29 题

HMMT February 2008 — Guts Round — Problem 29

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 15 ] Let ( x, y ) be a pair of real numbers satisfying − y x 56 x + 33 y = , and 33 x − 56 y = . 2 2 2 2 x + y x + y Determine the value of | x | + | y | . ◦
解析
  1. [ 15 ] Let ( x, y ) be a pair of real numbers satisfying − y x 56 x + 33 y = , and 33 x − 56 y = . 2 2 2 2 x + y x + y Determine the value of | x | + | y | . 11 Answer: Observe that 65 1 x − yi = = 33 x − 56 y + (56 x + 33 y ) i = (33 + 56 i )( x + yi ) . 2 2 x + yi x + y So ( ) 2 1 1 7 − 4 i 2 ( x + yi ) = = = . 2 33 + 56 i (7 + 4 i ) 65 ( ) 7 4 It follows that ( x, y ) = ± , − . 65 65 ◦