HMMT 十一月 2016 · 团队赛 · 第 3 题

英文解析

3. [ 3 ] Complex number ω satisfies ω = 2. Find the sum of all possible values of
   4 3 2
   ω + ω + ω + ω + 1 .
   Proposed by: Henrik Boecken
   Answer: 5
   ω − 1 15
   4 3 2
   The value of ω + ω + ω + ω + 1 = = . The sum of these values is therefore the sum ofω − 1 ω − 1
   5 5 4 31
   over the five roots ω . Substituting z = ω − 1, we have that ( z + 1) = 2, so z + 5 z + 10 z +
   ω − 1
   10 z + 5 z − 1 = 0. The sum of the reciprocals of the roots of this equation is − = 5 by Vieta’s.25 − 1