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HMMT 二月 2002 · 冲刺赛 · 第 33 题

HMMT February 2002 — Guts Round — Problem 33

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [8] The expression b x c denotes the greatest integer less than or equal to x . Find the value of
    ⌊ ⌋
    2002!
    .
    2001! + 2000! + 1999! + · · · + 1!
    √
解析

英文解析

  1. The expression b x c denotes the greatest integer less than or equal to x . Find the value of
    ⌊ ⌋
    2002!
    .
    2001! + 2000! + 1999! + · · · + 1!
    Solution: 2000 We break up 2002! = 2002(2001)! as
    2000(2001!) + 2 · 2001(2000!) = 2000(2001!) + 2000(2000!) + 2002 · 2000(1999!) > 2000(2001! + 2000! + 1999! + · · · + 1!) .
    On the other hand,
    2001(2001! + 2000! + · · · + 1!) > 2001(2001! + 2000!) = 2001(2001!) + 2001! = 2002! .
    Thus we have 2000 < 2002! / (2001! + · · · + 1!) < 2001, so the answer is 2000.
    √8