HMMT 二月 2004 · 冲刺赛 · 第 28 题
HMMT February 2004 — Guts Round — Problem 28
题目详情
- [10] Find the value of ( ) ( ) ( ) ( ) 2003 2003 2003 2003
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- · · · + . 1 4 7 2002
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解析
- Find the value of ( ) ( ) ( ) ( ) 2003 2003 2003 2003
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- · · · + . 1 4 7 2002 2003 Solution: (2 − 2) / 3 √ Let ω = − 1 / 2+ i 3 / 2 be a complex cube root of unity. Then, by the binomial theorem, we have ( ) ( ) ( ) ( ) 2003 2003 2003 2003 2 2003 2 3 4 2005 ω ( ω + 1) = ω + ω + ω + · · · + ω 0 1 2 2003 ( ) ( ) ( ) ( ) 2003 2003 2003 2003 2003 2 = + + + · · · + 0 1 2 2003 ( ) ( ) ( ) ( ) 2003 2003 2003 2003 − 2 − 1 2003 − 2 − 3 − 4 − 2005 ω ( ω + 1) = ω + ω + ω + · · · + ω 0 1 2 2003 ( ) 2003 If we add these together, then the terms for n ≡ 1 (mod 3) appear with coefficient n 2 3, while the remaining terms appear with coefficient 1 + ω + ω = 0. Thus the desired 2 2003 2003 − 2 − 1 2003 2 sum is just ( ω ( ω + 1) + 2 + ω ( ω + 1) ) / 3. Simplifying using ω + 1 = − ω − 1 2003 2003 and ω + 1 = − ω gives ( − 1 + 2 + − 1) / 3 = (2 − 2) / 3.
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