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PUMaC 2013 · 团队赛 · 第 13 题

PUMaC 2013 — Team Round — Problem 13

专题
Discrete Math / 离散数学
难度
L3
来源
PUMaC

题目详情

  1. The equation x − 2 x − 1 = 0 has five complex roots r , r , r , r , r . Find the value of 1 2 3 4 5 1 1 1 1 1
        • . 8 8 8 8 8 r r r r r 1 2 3 4 5
解析
  1. The equation x − 2 x − 1 = 0 has five complex roots r , r , r , r , r . Find the value of 1 2 3 4 5 1 1 1 1 1
        • . 8 8 8 8 8 r r r r r 1 2 3 4 5 SOLUTION: By Viete’s formulae, we know ∑ r = 2 i and ∑ r r = 0 . i j Rearranging the original equation, we get − 8 2 x = ( x − 2) and hence ∑ ∑ − 8 2 r = ( r − 2) i i ∑ ∑ 2 = r − 4 r + 20 i i ∑ ∑ ∑ 2 = ( r ) − r r − 4 r + 20 i i j i = 16 . ANSWER: 16 4