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HMMT 二月 2018 · 团队赛 · 第 2 题

HMMT February 2018 — Team Round — Problem 2

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 25 ] Is the number ( ) ( ) ( ) ( ) 1 1 1 1 1 + 1 + 1 + . . . 1 + 2 4 6 2018 greater than, less than, or equal to 50? n
解析
  1. [ 25 ] Is the number ( ) ( ) ( ) ( ) 1 1 1 1 1 + 1 + 1 + . . . 1 + 2 4 6 2018 greater than, less than, or equal to 50? Proposed by: Henrik Boecken Call the expression S . Note that ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 1 1 + 1 + 1 + . . . 1 + < 1 + 1 + 1 + . . . 1 + 2 4 6 2018 1 3 5 2017 Multiplying these two products together, we get ( ) ( ) ( ) ( ) 1 1 1 1 1 + 1 + 1 + . . . 1 + 1 2 3 2018 2 3 4 2019 = · · . . . 1 2 3 2018 = 2019 This shows that √ 2 S < 2019 = ⇒ S < 2019 < 50 as desired. n