HMMT 二月 2007 · CALC 赛 · 第 8 题
HMMT February 2007 — CALC Round — Problem 8
题目详情
- [ 6 ] Suppose that ω is a primitive 2007 root of unity. Find 2 − 1 . j 2 − ω j =1 k For this problem only, you may express your answer in the form m · n + p, where m, n, k, and p are th n positive integers. Note that a number z is a primitive n root of unity if z = 1 and n is the smallest k number amongst k = 1 , 2 , . . . , n such that z = 1.
解析
- [ 6 ] Suppose that ω is a primitive 2007 root of unity. Find 2 − 1 . j 2 − ω j =1 k For this problem only, you may express your answer in the form m · n + p, where m, n, k, and p are th n positive integers. Note that a number z is a primitive n root of unity if z = 1 and n is the smallest k number amongst k = 1 , 2 , . . . , n such that z = 1. 2006 Answer: 2005 · 2 + 1 . Note that ∑ ∏ 2006 i ( z − ω ) 1 1 j =1 i 6 = j
- · · · + = 2006 2006 z − ω z − ω ( z − ω ) · · · ( z − ω ) [ ] d 2006 2005 2005 2004 z + z + · · · + 1 2006 z + 2005 z + · · · + 1 z − 1 d z = = · 2006 2005 2006 2005 z + z + · · · + 1 z + z + · · · + 1 z − 1 2006 2005 2004 2007 2006 2006 z − z − z − · · · − 1 z − 1 2006 z − 2007 z + 1 = · = . 2007 2007 z − 1 z − 1 ( z − 1)( z − 1) 2006 2005 · 2 +1 Plugging in z = 2 gives ; whence the answer. 2007 2 − 1