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HMMT 二月 2009 · 代数 · 第 5 题

HMMT February 2009 — Algebra — Problem 5

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 4 ] Let a , b , and c be the 3 roots of x − x + 1 = 0. Find + + . a +1 b +1 c +1 π
解析
  1. [ 4 ] Let a , b , and c be the 3 roots of x − x + 1 = 0. Find + + . a +1 b +1 c +1 Answer: − 2 Solution: We can substitute x = y − 1 to obtain a polynomial having roots a + 1, b + 1, c + 1, namely, 3 3 2 ( y − 1) − ( y − 1) + 1 = y − 3 y + 2 y + 1. The sum of the reciprocals of the roots of this polynomial 2 is, by Viete’s formulas, = − 2. − 1 π