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HMMT 二月 2001 · 团队赛 · 第 7 题

HMMT February 2001 — Team Round — Problem 7

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. The Fibonacci numbers are defined by F = F = 1 and F = F + F for n ≥ 1.
    1 2 n +2 n +1 n
    The Lucas numbers are defined by L = 1, L = 2, and L = L + L for n ≥ 1.
    1 2 n +2 n +1 n
    ∏15
    2 n fn =1Fn
    Calculate .
    ∏13
    n =1 n lsin 10+sin 20+sin 30+sin 40+sin 50+sin 60+sin 70+sin 80
解析

英文解析

  1. The Fibonacci numbers are defined by F = F = 1 and F = F + F for n ≥ 1.
    1 2 n +2 n +1 n
    The Lucas numbers are defined by L = 1, L = 2, and L = L + L for n ≥ 1.
    1 2 n +2 n +1 n
    ∏15
    2 n fn =1Fn
    Calculate .
    ∏13
    n =1 n l
    2 n f
    Solution: It is easy to show that L = , so the product above is L 4 L 5 = 843 ·
    n 1 1
    1364 = 1149852 .n fsin 10+sin 20+sin 30+sin 40+sin 50+sin 60+sin 70+sin 80